8.EE.C.8Common CoreMathExpressions and EquationsGrade 8
8.EE.C.8: Solving Systems of Two Linear Equations
In plain English: 8.EE.C.8 is the Common Core grade 8 math standard that asks students to analyze and solve pairs of simultaneous linear equations. The solution of a system is the point where the two lines cross, because it makes both equations true. Students solve systems by graphing, by algebra and by inspection, and use them in real-world problems.
Analyze and solve pairs of simultaneous linear equations.
a.Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b.Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c.Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations. Also written as 8.EE.8 · Official standard
Students meet systems of two linear equations, two equations in x and y that must be true at the same time. The central idea is that the solution of a system is the point where the two graphs cross: every point on a line makes its equation true, so the point on both lines makes both equations true.
Students then solve systems three ways. They graph both lines and estimate the crossing point, which may fall between grid lines. They solve algebraically, by substitution (replacing one variable with an expression equal to it) or elimination (adding or subtracting the equations so that one variable cancels), which gives the exact answer. And they solve simple cases by inspection, for example by seeing that 3x + 2y cannot be 5 and 6 at the same time. The lesson ends with real-world problems and with deciding whether two lines through given points intersect.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why the point where two lines cross is the solution of the system, because it makes both equations true
Graph two linear equations and estimate the solution of the system
Solve a system of two linear equations algebraically, by substitution or elimination
Solve simple systems by inspection, including systems with no solution or infinitely many solutions
Write and solve a system of two linear equations for a real-world or mathematical problem
Prior Knowledge Required
Students should already be comfortable with:
Solving linear equations in one variable, including variables on both sides 8.EE.C.7
Finding the slope of a line and writing its equation as y = mx + b 8.EE.B.6
Plotting ordered pairs in all four quadrants 6.NS.C.8
Checking whether a number makes an equation true by substituting it 6.EE.B.5
Read the riddle aloud and give pairs three minutes. Ask them to write their lists in two columns.
Warm-Up Prompt
"I am thinking of two numbers. They add up to 10, and one is 4 more than the other. What are they?" Then: "List three pairs of numbers that add up to 10. List three pairs where the first is 4 more than the second. Which pair is on both lists?"
Pairs soon find 7 and 3. Write the two conditions as equations, x + y = 10 and x - y = 4, and point out that many pairs make the first one true and many pairs make the second one true, but only one pair makes both true at the same time. Tell students that a pair of equations like this is called a system, and today they will find the pair that works for both by graphing, by algebra and by just looking.
Direct Instruction20 minutes
Define the key words and write each one on an anchor chart:
System of linear equations: two linear equations with the same two variables, x and y, that we want to be true together. Simultaneous means "at the same time."
Solution of a system: an ordered pair (x, y) that makes both equations true. A pair that works in only one equation is not a solution.
Point of intersection: the point where two graphs cross. Every point on a line makes that line's equation true, so the point on both lines makes both equations true. That is why the intersection point is the solution.
Recall from 8.EE.B.6: in y = mx + b, m is the slope (the steepness, rise over run) and b is the y-intercept (where the line crosses the y-axis). Use them to graph each line quickly.
Solving algebraically: in the substitution method, replace one variable with an expression that equals it from the other equation. In the elimination method, add or subtract the equations so that one variable cancels. Both leave one equation in one variable, which students already know how to solve (8.EE.C.7).
Solving by inspection: looking at the equations and seeing the answer without working it through. Same slope and different y-intercepts: parallel lines, no solution. The same line written two ways: infinitely many solutions.
Work through the examples below. For each one, ask students to predict first: "Will these lines cross once, never, or everywhere?"
The solution is the point of intersection
Graph y = x + 1 and y = -2x + 10 on the same grid. Where do the lines cross, and why is that point the solution?
Equation: The lines cross at (3, 4). Check: 4 = 3 + 1 and 4 = -2(3) + 10, so (3, 4) makes both equations true. The point (5, 6) is on the first line only, and (1, 8) is on the second line only, so neither one is a solution of the system.
Solving by substitution
Solve y = 4x - 1 and 3x + y = 13.
Equation: The first equation says y is the same as 4x - 1, so replace y in the second: 3x + (4x - 1) = 13. Then 7x - 1 = 13, 7x = 14 and x = 2. Substitute back: y = 4(2) - 1 = 7. Solution (2, 7). Check: 3(2) + 7 = 13.
Estimate by graphing, then solve by elimination
Solve x + y = 5 and 2x - y = 3. First estimate from a graph.
Equation: On a graph the lines cross near (2.7, 2.3). Add the equations to eliminate y: 3x = 8, so x = 8/3. Then y = 5 - 8/3 = 7/3. The exact solution is (8/3, 7/3), about (2.67, 2.33).
Solving by inspection, the official example
How many solutions do 3x + 2y = 5 and 3x + 2y = 6 have?
Equation: None. The same expression, 3x + 2y, cannot be 5 and 6 at the same time. On a graph, the two lines are parallel: both have slope -3/2, with different y-intercepts.
Do two lines through given points intersect? (official example)
Line 1 passes through (0, 1) and (2, 5). Line 2 passes through (1, -1) and (3, 3). Do the lines intersect?
Equation: Slope of line 1: (5 - 1) ÷ (2 - 0) = 2, so y = 2x + 1. Slope of line 2: (3 - (-1)) ÷ (3 - 1) = 2, so y = 2x - 3. Same slope, different y-intercepts: the lines are parallel and never intersect. The system y = 2x + 1, y = 2x - 3 has no solution.
Use Diagram 1 with Example 1: test the three marked points in both equations, and let students see that only the crossing point passes both tests. Use Diagram 2 with Example 3. The graph gives an estimate (a close value, not an exact one) because the lines cross between grid lines; algebra gives the exact answer. Remind students that a graph drawn by hand can be off by a few tenths, so a graph answer should always be checked in both equations.
Guided Practice15 minutes
Pairs work on four problems. After each one, a pair explains its method at the board and checks the answer in both equations.
Graph y = x - 1 and y = -x + 5 on the same grid and read the solution. ((3, 2): 2 = 3 - 1 and 2 = -3 + 5.)
Solve by substitution: x = 2y and x + 3y = 15. (2y + 3y = 15, so y = 3 and x = 6: (6, 3).)
Solve by elimination: 2x + y = 9 and 3x - y = 11. (Add: 5x = 20, so x = 4 and y = 1: (4, 1).)
At a bake sale, cookies cost $1 and brownies cost $2. Maria sold 30 items and collected $48. How many of each did she sell? (c + b = 30 and c + 2b = 48. Subtract: b = 18, so c = 12: 12 cookies and 18 brownies.)
Listen for students who find x and stop, and for students who check the answer in only one equation.
Independent Practice15 minutes
Students work alone, then compare with a partner.
By inspection: how many solutions do y = 3x + 2 and y = 3x - 5 have? (None: same slope, different y-intercepts, so the lines are parallel.)
By inspection: how many solutions do x + y = 12 and 2x + 2y = 24 have? (Infinitely many: the second equation is the first one doubled, so both give the same line.)
Solve y = -2x + 8 and y = x + 2. ((2, 4).)
A candle 30 cm tall burns down 2 cm per hour. A second candle 24 cm tall burns down 1 cm per hour. Both are lit at the same time. When are they the same height, and what is that height? (h = 30 - 2t and h = 24 - t: after 6 hours, both 18 cm tall.)
Line 1 passes through (0, 2) and (4, 4). Line 2 passes through (0, 5) and (2, 4). Do the lines intersect? (Yes. The slopes are 1/2 and -1/2, so y = 0.5x + 2 and y = -0.5x + 5 cross at (3, 3.5).)
Closure5-10 minutes
Exit ticket: (1) Is (2, -1) a solution of the system x + y = 1 and 3x + y = 5? Show your check. (Yes: 2 + (-1) = 1 and 3(2) + (-1) = 5.) (2) By inspection, solve y = -x + 4 and y = 2x + 4. (Both lines have y-intercept 4 and different slopes, so they cross only at (0, 4).) (3) Finish the sentence: "The point where two lines cross is the solution of the system because ..."
Differentiation Strategies
For Struggling Students
Give pre-drawn axes with the scale already marked, so students can focus on the lines and the crossing point
Start with systems where one equation is already solved for y, so substitution is one step
Use a two-column check table (equation 1, equation 2) for every answer, so students always test both equations
For Advanced Students
Ask for a system whose solution is (-1, 4) and whose lines have slopes 2 and -3, then a second system with the same solution written in the form ax + by = c
Ask: for which value of k does y = kx + 1 and y = 4x - 2 have no solution? (k = 4)
Extension (beyond this standard): preview HSA.REI.C.6 with a system that needs both equations multiplied before elimination, such as 3x + 4y = 10 and 2x - 3y = 1
Assessment Guidance
What to Look For
Every answer should be an ordered pair checked in both equations. Watch for students who stop after finding x, who check only one equation, or who lose a negative sign when they substitute an expression such as -5x + 2. On graphs, look for a consistent scale and for honest estimates ("about (2.7, 2.3)") when the crossing point falls between grid lines. When the variables cancel, a strong answer names the statement (0 = 0 or 0 = 5) and what it says about the lines.
02
Classroom Activities
3 Activities
1
Graph It, Check It
20 minPairs
Pairs graph six systems on grid paper, one system per grid, and read each solution from the graph. Then they check the point in both equations. When the lines cross between grid lines, they write an estimate and then solve the system algebraically to get the exact answer.
The 6 System Cards (Answer Key for the Teacher)
Card A: y = x + 2 and y = -x + 6 ((2, 4))
Card B: y = 2x and y = -x + 9 ((3, 6))
Card C: y = 0.5x + 1 and y = -x + 7 ((4, 3))
Card D: y = 3x - 4 and y = x + 1 (between grid lines, near (2.5, 3.5); exactly (2.5, 3.5))
Card E: y = -2x + 5 and y = x - 3 (between grid lines, near (2.7, -0.3); exactly (8/3, -1/3))
Card F: y = x + 3 and y = x - 2 (parallel lines, no solution)
Procedure
Use a ruler and a scale of one grid square per unit on both axes
Graph each line from its y-intercept and slope, and label it with its equation
Circle the crossing point and write its coordinates, or write "about (..., ...)" if it falls between grid lines
Check every answer by substituting it into both equations
Discussion Questions
For which two cards could you only estimate the solution? How did algebra help?
Card F never gave a crossing point. What did you notice about the two equations before you graphed them?
On Card D, one student read (2, 3) from the graph. How would checking in both equations show that it is wrong?
Modification for Distance Learning
Use a free online graphing tool. Pairs type both equations, click the crossing point to read its coordinates, and then check the point by hand in both equations.
2
Inspection Speed Round
15 minPairs
Project eight systems, one at a time. Pairs have 20 seconds to hold up a card that says "One solution," "No solution" or "Infinitely many," and, for one solution, to write the point on a mini whiteboard. No pencil work is allowed for the first answer: the point is to solve simple cases by inspection.
The 8 Rounds (Answer Key for the Teacher)
Round 1: y = 4 and x + y = 10 ((6, 4))
Round 2: 2x + 3y = 8 and 2x + 3y = 1 (no solution)
Round 3: x + 2y = 6 and 3x + 6y = 18 (infinitely many solutions)
Round 4: y = 2x + 7 and y = -3x + 7 ((0, 7))
Round 5: x = -2 and y = 3x ((-2, -6))
Round 6: y = -4x + 1 and y = -4x + 3 (no solution)
Round 7: 4x - y = 2 and 8x - 2y = 4 (infinitely many solutions)
Round 8: x - y = 0 and x + y = 12 ((6, 6))
Procedure
Show one system. Pairs answer within 20 seconds
Ask one pair to say what they saw: a fixed value of x or y, the same y-intercept, the same left side, or one equation that is a multiple of the other
Then everyone checks the answer by substituting or by graphing quickly
Discussion Questions
Rounds 2 and 6 both had no solution. What clue did the two systems share?
In Rounds 3 and 7, how could you tell that the two equations give the same line?
Round 4 was solved without any arithmetic. What made (0, 7) easy to see?
3
Fundraiser Stations
20 minGroups of 3-4
The class is planning a fundraiser. Groups rotate through three stations, every 6 minutes. At each station they name two variables, write a system of two equations, solve it with any method, and answer the question in a full sentence with units.
The Three Stations
Station 1, T-shirts: Company A charges a $50 setup fee plus $6 per shirt. Company B charges a $20 setup fee plus $8 per shirt. For how many shirts do the two companies charge the same, and which company is cheaper for the club's order of 40 shirts?
Station 2, Pizza sale: The club sold 45 pizzas. Small pizzas cost $8 and large pizzas cost $12. The club collected $440. How many of each size did it sell?
Station 3, Class plants: A bean plant is 4 cm tall and grows 1.5 cm per week. A pea plant is 10 cm tall and grows 0.5 cm per week. When are the plants the same height, and how tall are they then?
Answer Key for the Teacher
Station 1: c = 50 + 6n and c = 20 + 8n. The costs match at 15 shirts ($140). For 40 shirts, A costs $290 and B costs $340, so A is cheaper.
Station 2: s + l = 45 and 8s + 12l = 440. Multiply the first equation by 8 and subtract: 4l = 80, so l = 20 and s = 25.
Station 3: h = 4 + 1.5w and h = 10 + 0.5w. Then w = 6: after 6 weeks both plants are 13 cm tall.
Procedure
At each station, write "Let ... = ..." for both variables before writing any equation
Solve the system and check the answer in both equations and in the story
Write one sentence that answers the question, with units
Challenge Variation
At the T-shirt station, graph both cost equations for 0 to 40 shirts. Shade the part of the graph where Company B is cheaper, and explain what the crossing point means for the club.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Solution Is the Point on Both Lines
The lines y = x + 1 (solid) and y = -2x + 10 (dashed), drawn to scale, cross at (3, 4). The point (5, 6) is on the first line only and (1, 8) is on the second line only, so each one fails one of the equations. Only (3, 4) makes both equations true, so it is the solution of the system.
Diagram 2: Estimate from the Graph, Then Solve Exactly
The lines for x + y = 5 and 2x - y = 3, drawn to scale, cross between grid lines, so the graph gives an estimate of about (2.7, 2.3). Adding the equations eliminates y and gives the exact solution (8/3, 7/3).
04
Homework Assignment
~30 min
8.EE.C.8 Homework: Systems of Linear Equations
Directions: Show your work. Use grid paper and a ruler for graphs, with one grid square per unit. Write every solution as an ordered pair and check it in both equations. For word problems, say what each variable stands for and answer in a sentence with units.
Part 1: Graphs and Solutions (Problems 1-2)
Graph y = -x + 4 and y = 3x - 4 on the same grid, with one grid square per unit. (a) Write the solution of the system and check it in both equations. (b) Is (1, 3) a solution of the system? Use your graph and the equations to explain.
Graph y = x + 5 and y = -2x + 12 on the same grid. (a) Estimate the solution from your graph to the nearest tenth. (b) Solve the system algebraically and compare the exact answer with your estimate.
Part 2: Algebra and Inspection (Problems 3-4)
Solve each system algebraically and check your answer: (a) y = 2x - 5 and 3x + 2y = 11 (b) 5x + 2y = 4 and 3x - 2y = 12
By inspection, decide whether each system has one solution, no solution or infinitely many solutions, and explain what you saw. If there is one solution, give it. (a) 5x - y = 3 and 5x - y = -1 (b) y = -3x + 2 and 6x + 2y = 4 (c) y = 8 and y = 2x
Part 3: Problems and Lines Through Points (Problems 5-6)
A class orders 36 sandwiches for a field trip. Turkey sandwiches cost $5 and veggie sandwiches cost $4. The order costs $164. Write a system of equations, solve it, and say how many of each kind the class ordered.
Line A passes through (0, -1) and (2, 3). Line B passes through (0, 5) and (3, -1). (a) Find the slope and the equation of each line. (b) Do the lines intersect? If so, find the point and check it in both equations. (c) Line C passes through (1, 4) and (3, 8). Does Line C intersect Line A? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Graphs
Both lines graphed to scale and labeled; crossing point read or estimated well
Small plotting errors or no labels
Graphs missing or wrong
Algebraic Solutions
Both values found correctly and written as an ordered pair
One value wrong, or only x found
Method missing or incorrect
Checks and Explanations
Every answer checked in both equations; inspection answers explained
Checks in one equation only, or vague reasons
No checks or reasons
Word Problems
Variables defined, system correct, answer in a sentence with units
System correct but answer incomplete
System missing or wrong
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
The graphs of two linear equations cross at the point (-2, 5). What does this tell you?
Answer: A
A point of intersection lies on both lines, so its coordinates make both equations true at the same time. That is why it is the solution of the system. Choice B splits the ordered pair between the two equations. Choice C treats the two coordinates as two separate solutions, but (-2, 5) is one solution. Choice D forgets that a crossing point is on both lines.
Question 2 of 20 · Multiple Choice
Which ordered pair is the solution of y = 3x - 1 and x + y = 15?
Answer: C
Substitute: x + (3x - 1) = 15, so 4x = 16, x = 4 and y = 11. Check: 11 = 3(4) - 1 and 4 + 11 = 15. Choice A works in x + y = 15 only, and choice B works in y = 3x - 1 only. Choice D swaps the coordinates: it fails y = 3x - 1, since 4 is not 3(11) - 1.
Question 3 of 20 · Multiple Choice
Solve by substitution: y = x - 4 and 3x + y = 16.
Answer: B
Replace y with x - 4: 3x + x - 4 = 16, so 4x = 20 and x = 5. Then y = 5 - 4 = 1. Choice A substitutes x + 4 instead of x - 4 (4x + 4 = 16, so x = 3) and then uses y = x - 4 to get y = -1. Choice C swaps the coordinates. Choice D finds x = 5 but then adds 4 instead of subtracting it.
Question 4 of 20 · Multiple Choice
Solve by elimination: 3x + 2y = 16 and 5x - 2y = 16.
Answer: D
Add the equations: 8x = 32, so x = 4. Then 3(4) + 2y = 16, so 2y = 4 and y = 2. Check in the second equation: 20 - 4 = 16. Choice A substitutes x = 4 into 5x - 2y = 16 but solves 20 + 2y = 16, a sign error. Choice B swaps the coordinates. Choice C adds 12 to 16 instead of subtracting it: 2y = 28, so y = 14.
Question 5 of 20 · Multiple Choice
How many solutions does the system 4x - y = 7 and 4x - y = 2 have?
Answer: A
The same expression, 4x - y, cannot equal 7 and 2 at the same time, so no pair works in both. The graphs are parallel lines (both have slope 4). Choice C would need the two equations to be the same line. Choice D is impossible for two lines: two different straight lines can cross at most once.
Question 6 of 20 · Multiple Choice
Which system has infinitely many solutions?
Answer: B
In choice B, the second equation is the first one multiplied by 3, so both describe the same line and every point on it is a solution. Choice A has the same slope and different y-intercepts, so the lines are parallel: no solution. Choices C and D have one solution each, (4, 0) and (0, 0); a student who sees a repeated number such as 4 may pick C by mistake.
Question 7 of 20 · Multiple Choice
Solve by inspection: x = 4 and x + y = 13.
Answer: C
The first equation already gives x = 4. Then 4 + y = 13, so y = 9, and the solution is (4, 9). Choice A swaps the coordinates. Choice B uses the total 13 as y. Choice D adds 4 to 13 instead of subtracting.
Question 8 of 20 · Multiple Choice
The lines y = -x + 8 and y = 2x - 3 cross between grid lines. Which gives a good estimate from the graph and the exact solution?
Answer: D
Set the two expressions for y equal: 2x - 3 = -x + 8, so 3x = 11 and x = 11/3, about 3.7. Then y = -11/3 + 8 = 13/3, about 4.3. Choice A swaps x and y. Choice B subtracts 3 from 8 instead of adding it (3x = 5). Choice C rounds to the nearest grid point, (4, 4), which is on y = -x + 8 but not on y = 2x - 3.
Question 9 of 20 · Multiple Choice
A museum charges $12 per adult and $7 per child. A group of 9 people paid $83. How many adults were in the group?
Answer: B
a + c = 9 and 12a + 7c = 83. Substitute c = 9 - a: 12a + 63 - 7a = 83, so 5a = 20 and a = 4. There were 4 adults and 5 children: 4($12) + 5($7) = $83. Choice A is the number of children. Choice C divides 83 by 12 and rounds. Choice D stops at 5a = 20 and reports 20.
Question 10 of 20 · Multiple Choice
Line P passes through (0, 4) and (2, 0). Line Q passes through (0, -2) and (3, 4). Do the lines intersect?
Answer: A
Line P has slope (0 - 4) ÷ (2 - 0) = -2, so y = -2x + 4. Line Q has slope (4 - (-2)) ÷ (3 - 0) = 2, so y = 2x - 2. The slopes differ, so the lines cross: -2x + 4 = 2x - 2 gives x = 1.5 and y = 1. Choice B drops the negative sign on the slope of P and thinks both slopes are 2. Choice C swaps the coordinates. Choice D is a given point on line P only.
Question 11 of 20 · Multiple Choice
Solve x = 3y + 1 and 2x - 5y = 4.
Answer: C
Substitute: 2(3y + 1) - 5y = 4, so 6y + 2 - 5y = 4 and y = 2. Then x = 3(2) + 1 = 7. Check: 14 - 10 = 4. Choice A swaps the coordinates. Choice B works in x = 3y + 1 but not in 2x - 5y = 4, since 8 - 5 = 3. Choice D multiplies only 3y by 2 (6y + 1 - 5y = 4, so y = 3 and x = 10).
Question 12 of 20 · Multiple Choice
Leah has $40 and saves $5 each week. Omar has $10 and saves $8 each week. After how many weeks will they have the same amount, and how much will each have?
Answer: D
m = 40 + 5w and m = 10 + 8w. Then 40 + 5w = 10 + 8w, so 30 = 3w and w = 10. Each has 40 + 5(10) = $90. Choice A stops at 3w = 30 and uses 30 as the number of weeks. Choice B divides 30 by Leah's rate of 5 instead of by the difference in rates, 3. Choice C finds 10 weeks but uses Omar's rate with Leah's start (40 + 80).
Question 13 of 20 · Multiple Choice
Which point is on the graph of y = 3x - 2 but is NOT a solution of the system y = 3x - 2 and y = x + 4?
Answer: B
(1, 1) is on the first line, since 3(1) - 2 = 1, but not on the second, since 1 + 4 = 5. The solution of the system is (3, 7), which is on both lines, so choice A is a solution. Choices C and D are on y = x + 4 but not on y = 3x - 2; a student who tests the wrong equation may pick them.
Question 14 of 20 · Multiple Choice
Tara solves 2x - 3y = 6 and -4x + 6y = -12. She multiplies the first equation by 2 and adds the result to the second. What does she get, and what does it mean?
Answer: C
Doubling gives 4x - 6y = 12. Adding -4x + 6y = -12 gives 0 = 0, a true statement: the equations describe the same line, so every point on it is a solution. Choice B adds 12 and 12 instead of 12 and -12. Choice A reads the zeros as coordinates. Choice D confuses a true statement with a false one.
Question 15 of 20 · Short Answer
Solve by substitution and check: y = -3x + 1 and 2x - y = 14.
Replace y: 2x - (-3x + 1) = 14, so 5x - 1 = 14, 5x = 15 and x = 3. Then y = -3(3) + 1 = -8. Solution: (3, -8). Check: -8 = -9 + 1 and 2(3) - (-8) = 14. Watch for students who write 2x + 3x + 1: the minus sign must go to both terms.
Question 16 of 20 · Short Answer
Solve by elimination: 2x + 3y = 4 and 4x - 3y = 26.
Add the equations to eliminate y: 6x = 30, so x = 5. Then 2(5) + 3y = 4, so 3y = -6 and y = -2. Solution: (5, -2). Check: 4(5) - 3(-2) = 20 + 6 = 26.
Question 17 of 20 · Short Answer
Graph y = x + 2 and y = -0.5x + 8 on the same grid. Read the solution from your graph, then solve algebraically. In one sentence, explain why the crossing point is the solution of the system.
The lines cross at (4, 6). Algebra: x + 2 = -0.5x + 8, so 1.5x = 6 and x = 4; y = 4 + 2 = 6. The point is on both lines, so it makes both equations true at the same time, which is what a solution of a system means.
Question 18 of 20 · Short Answer
Two school groups rent vans and buses. Group A uses 5 vans and 2 buses to carry 130 students. Group B uses 3 vans and 4 buses to carry 190 students. Every van holds the same number of students, and so does every bus. How many students does one van hold, and how many does one bus hold?
Let v and b be the students per van and per bus. 5v + 2b = 130 and 3v + 4b = 190. Double the first equation: 10v + 4b = 260. Subtract the second: 7v = 70, so v = 10. Then 5(10) + 2b = 130, so b = 40. A van holds 10 students and a bus holds 40. Check: 3(10) + 4(40) = 190.
Question 19 of 20 · Short Answer
Without solving, explain why the system 6x + 4y = 10 and 3x + 2y = 7 has no solution. What do the graphs look like?
Divide the first equation by 2: 3x + 2y = 5. The expression 3x + 2y cannot be 5 and 7 at the same time, so no pair (x, y) works in both equations. The graphs are parallel lines (both have slope -3/2) with different y-intercepts.
Question 20 of 20 · Short Answer
Tank A holds 800 liters of water and is drained at 40 liters per minute. Tank B holds 200 liters and is filled at 20 liters per minute. Both start at the same time. Write a system of equations, solve it, and explain what the solution means.
Let t be the minutes and w the liters. w = 800 - 40t and w = 200 + 20t. Then 800 - 40t = 200 + 20t, so 600 = 60t and t = 10. Both tanks hold 800 - 400 = 400 liters. After 10 minutes, both tanks hold 400 liters.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.C.8 mean?
8.EE.C.8 means students can analyze and solve a system of two linear equations in two variables. They learn that the solution is the point where the two graphs cross, because that point makes both equations true. They solve systems by graphing, by algebra and by inspection, and they use systems to solve real-world problems.
Is 8.EE.C.8 taught in grade 8 or in Algebra I?
8.EE.C.8 is a grade 8 standard, usually taught near the end of the linear equations unit in Grade 8 Math. Algebra I returns to systems in HSA.REI.C.6, with harder coefficients (the numbers multiplied by the variables), and in HSA.CED.A.3, where systems describe constraints in modeling problems.
Why is the point of intersection the solution of a system?
Because it is the only point on both lines. Every point on a line makes that line's equation true. A point on both lines makes both equations true at the same time, and that is the definition of a solution of a system. A point on only one line fails the other equation.
What does "solve by inspection" mean in 8.EE.C.8?
It means finding the answer by looking closely at the equations, without long calculations. Examples: y = 9 and x + y = 14 gives x = 5 right away; y = 5x + 1 and y = 5x - 4 have the same slope and different y-intercepts, so there is no solution; 2x - y = 3 and 4x - 2y = 6 are the same line, so there are infinitely many solutions.
How can a system have no solution or infinitely many solutions?
Two lines in a plane can cross once, never or everywhere. Parallel lines (same slope, different y-intercepts) never meet, so the system has no solution. Two equations that describe the same line share every point, so the system has infinitely many solutions. When solving algebraically, the first case ends with a false statement such as 0 = 5, and the second with a true one such as 0 = 0.
Should students use substitution or elimination?
Either one. The standard says "algebraically," and both methods count. Substitution is quick when one equation is already solved for x or y, such as y = 3x - 2. Elimination is quick when a variable has the same or opposite coefficients in both equations, such as 2y and -2y. Students should be able to use both and choose.
Why estimate from a graph if algebra gives the exact answer?
A graph shows what the answer means and helps catch mistakes. If algebra gives (40, -3) but the lines clearly cross near (6, 1), something went wrong. The standard asks students to estimate solutions by graphing because many crossing points fall between grid lines, where a graph can only give an approximate answer.
How do you tell whether two lines through given points intersect?
Find the slope of each line from its two points. Different slopes mean the lines cross exactly once. The same slope means they are parallel or the same line: check whether a point of one line is on the other. For example, the line through (0, 0) and (1, 3) and the line through (0, 2) and (2, 8) both have slope 3, and (0, 2) is not on y = 3x, so they never meet.
What mistakes do students make with systems of equations?
A common one is finding x and stopping, without finding y. Another is checking the answer in only one equation, which cannot tell a solution from a point on one line. Students also lose negative signs when they substitute an expression such as -4x + 9, and they swap the coordinates when they write the ordered pair.
How does 8.EE.C.8 connect to the rest of math?
It uses 8.EE.C.7, since every system turns into a one-variable equation, and it uses slope and y-intercept from 8.EE.B.6. In high school, HSA.REI.C.6 extends the same methods, and HSA.REI.D.11 uses intersections of graphs to solve other kinds of equations. Systems also appear in science and money problems whenever two conditions must be true together.
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Related Standards
5 standards
These standards connect to 8.EE.C.8: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
