HSA.REI.D.10: Graphs as the Set of All Solutions of an Equation
In plain English: HSA.REI.D.10 is the Common Core algebra standard that asks students to understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, which could be a line. Every point on the graph makes the equation true, and every solution is a point on the graph. It is usually taught in Algebra I.
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Common Core State Standards for Mathematics · Domain: Reasoning with Equations and Inequalities (REI) · Cluster: Represent and solve equations and inequalities graphically Also written as HSA-REI.D.10 or A-REI.10 · Official standard
In this lesson, students learn what a graph of an equation in two variables actually is: the set of every ordered pair (x, y) that makes the equation true, plotted in the coordinate plane. They test points by substitution, build tables of solutions, and see those solutions fill in a line, a parabola or a circle. The idea works in both directions: a point on the graph is a solution, and a solution is a point on the graph.
This understanding is what makes graphing a tool for reasoning later on. Students who know that every point on a curve satisfies its equation can explain why intersection points solve systems and why the x-intercepts of y = f(x) solve f(x) = 0.
Learning Objectives
By the end of this lesson, students will be able to:
Decide whether a point lies on the graph of an equation by substituting its coordinates
Generate solutions of an equation in two variables with a table and plot them to form the graph
Explain why the graph of an equation is a continuous curve (sometimes a line) when every real input gives a solution
Find a missing coordinate of a point that lies on a given graph
Distinguish the graph of an equation from the set of solutions that make sense in a context
Prior Knowledge Required
Students should already be comfortable with:
Plotting ordered pairs in all four quadrants of the coordinate plane
Knowing that the graph of a function is the set of its input-output pairs 8.F.A.1
Evaluating expressions with integers, fractions and exponents
Write the equation x + y = 6 on the board and pose the prompt.
Warm-Up Prompt
"Find as many pairs of numbers x and y as you can that add up to 6. Include at least one pair with a negative number and one with a fraction. Then plot all your pairs on one grid."
Have several students add their points to a class grid with sticky dots. The dots line up. Ask: "Could there be a solution that is not on this line? Is every point on this line a solution?" Test (2.5, 3.5) and (10, -4) together. This sets up the lesson's main idea: the line is the set of all solutions.
Direct Instruction20 minutes
State the definition: the graph of an equation in two variables is the set of all ordered pairs (x, y) that make the equation true. Then model three routines:
Test a point: substitute x and y into the equation. If both sides are equal, the point is on the graph; if not, it is off the graph.
Build the graph: choose x-values, compute the matching y-values, plot the pairs, and connect them when every real x in between also gives a solution.
Use a known point: if a point is on the graph, its coordinates satisfy the equation, so you can substitute and solve for a missing coordinate.
Testing points on a line
Are (3, 2) and (4, 1) on the graph of 2x + 3y = 12? Substitute: 2(3) + 3(2) = 12, but 2(4) + 3(1) = 11.
Equation: (3, 2) is on the graph; (4, 1) is not
Building a curve from a table
For y = x² - 1, the x-values -2, -1, 0, 1, 2 give the solutions (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3). Values in between also work: x = 1.5 gives y = 1.25.
Equation: The solutions form a parabola with vertex (0, -1)
A curve that is not a function
Is (3, 4) on the graph of x² + y² = 25? What about (2, 4)? Substitute: 9 + 16 = 25, but 4 + 16 = 20.
Equation: (3, 4) is on the circle of radius 5; (2, 4) is not
Finding a missing coordinate
The point (k, 7) is on the graph of y = 2x + 1, and the point (-2, m) is on the graph of y = x³ - 2x. Substitute each point into its equation.
Equation: 7 = 2k + 1, so k = 3; m = (-2)³ - 2(-2) = -4
Solutions in a context
Granola bars cost $2 and juice boxes cost $3. Spending exactly $24 means 2g + 3j = 24. The whole-number solutions are (0, 8), (3, 6), (6, 4), (9, 2) and (12, 0).
Equation: The graph of 2g + 3j = 24 is a line, but only these 5 points make sense for the purchase
Guided Practice15 minutes
Pairs work through a "sort the points" card: for each of the equations y = -2x + 5, y = |x| - 3 and x² + y² = 10, they get six points and sort them into "on the graph" and "off the graph" by substitution, then sketch the graph and check that the sort matches the picture. Sample points for y = |x| - 3: (-4, 1) is on, (2, -1) is on, (0, 3) is off. Circulate and watch for students who swap x and y when substituting, and for students who decide by "eyeballing" a sketch instead of substituting.
Independent Practice10-15 minutes
Students complete four tasks on their own: (1) make a table for y = x² - 4x with x = 0 to 4 and graph it, (2) test three points on the graph of 3x - y = 7, (3) find the missing coordinate for (a, 10) on y = 3x - 2, and (4) list all whole-number solutions of a two-variable budget equation and explain why its graph in context is a set of points rather than the full line.
Closure5-15 minutes
Exit ticket: "The point (5, 9) is on the graph of y = 2x - 1. Write two true sentences about this fact: one about the equation and one about the graph. Then name one point that is not on the graph and prove it." Look for sentences like "x = 5 and y = 9 make the equation true" and "(5, 9) lies on the line."
Differentiation Strategies
For Struggling Students
Provide a two-column substitution frame: "Left side = ___, right side = ___, equal? yes/no"
Start with linear equations already in the form y = mx + b before moving to x² + y² = 25
Use a partially completed table so students focus on plotting and connecting points
For Advanced Students
Find every point with integer coordinates on the graph of x² + y² = 65 and explain how they know the list is complete
Graph y = 1/x from a table and explain why the graph has no point with x = 0
Write an equation whose graph passes through three given points, such as (0, 1), (1, 2) and (-1, 2)
Assessment Guidance
What to Look For
Listen for students who can explain in both directions: a point on the graph makes the equation true, and a solution of the equation is a point on the graph. Students who only say "the line goes through it" without substituting have not yet connected the graph to the equation. Also check that students can explain why a context may allow only some points of a graph.
02
Classroom Activities
3 Activities
1
Human Graph
15-20 minWhole class
Tape a large coordinate grid on the floor (or use a grid projected on the board). Each student gets a card with an ordered pair. For each equation announced, students stand on the grid only if their point is a solution.
Procedure
Hand out cards with points such as (-2, -5), (0, -1), (1, 1), (2, 3), (3, 5), (-1, 0), (2, 0), (0, 2), (-2, 3) and (1, -1)
Announce y = 2x - 1: the students holding (-2, -5), (0, -1), (1, 1), (2, 3) and (3, 5) step forward, and the class sees a line
Announce y = x² - 1: now (-1, 0), (0, -1), (2, 3) and (-2, 3) step forward, and the class sees a parabola
Each student standing must justify the choice aloud by substituting
Discussion Questions
Why did (0, -1) and (2, 3) stand up for both equations?
If we had a student for every possible point, what would the class look like for each equation?
Modification for Distance Learning
Use a shared online graph where students each drop one point, then color it green or red after substituting.
2
Fill In the Gaps
20 minPairs
Pairs graph y = x² - 1 from integer x-values, then keep adding points at halves and quarters until they are convinced the points form an unbroken curve.
Procedure
Round 1: x = -2, -1, 0, 1, 2 (5 points)
Round 2: x = -1.5, -0.5, 0.5, 1.5 (for example, x = 0.5 gives y = -0.75)
Round 3: each partner picks two more x-values of their choice, including one they would find hard to compute by hand, and uses a calculator
Pairs write one sentence explaining why the final graph is drawn as a smooth curve instead of separate dots
Contrast Variation
Repeat with y = 12/x for x = 1, 2, 3, 4, 6, 12 and the negative values. Students notice that no point has x = 0, so the graph has two separate branches. The graph is still exactly the set of all solutions.
3
Budget Line
20 minGroups of 3
Groups model a real purchase with a two-variable equation and compare the graph of the equation with the points that make sense in context.
Scenario
A club sells T-shirts for $10 and hats for $5 and raises exactly $100: 10s + 5h = 100
Groups list every whole-number solution: s = 0, 1, 2, ..., 10 with h = 20 - 2s, which is 11 points
They graph the full line 10s + 5h = 100 in one color and the 11 sensible points in another
Discussion Questions
Is (2.5, 15) a solution of the equation? Is it a possible sale? (It is a solution, since 25 + 75 = 100, but half a T-shirt cannot be sold.)
What do the points on the axes, (10, 0) and (0, 20), mean?
How would the graph change if the goal were "at least $100"? (That leads to inequalities, HSA.REI.D.12.)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Points On and Off the Graph of 2x + 3y = 12
Drawn to scale. Every green point makes 2x + 3y = 12 true and lies on the line, for example 2(-3) + 3(6) = 12. The point (4, 1) gives 11, not 12, so it is not a solution and it sits just below the line.
Diagram 2: Solutions Filling In a Parabola and a Circle
Left: the table values for y = x² - 1 are plotted, and in-between values such as (1.5, 1.25) fill in the curve. Right: the graph of x² + y² = 25 is a circle of radius 5; it has 12 points with integer coordinates, but every point on the circle is a solution. Both panels are drawn to scale.
04
Homework Assignment
~30 min
HSA.REI.D.10 Homework: Is It on the Graph?
Directions: Show the substitution for every point you test, and write "on the graph" or "not on the graph" with a reason. When you graph, use a table of at least five points.
Part 1: Testing and Building Graphs (Problems 1-3)
Which of the points (2, 5), (-1, -4) and (0, -3) are on the graph of y = 4x - 3? Show your substitution for each.
Complete a table for y = x² - 2x using x = -1, 0, 1, 2, 3. Plot the points and describe the shape of the graph. Is (4, 6) on the graph?
Find every point on the graph of x² + y² = 169 that has x-coordinate 5. Explain why the graph of this equation is not the graph of a function, even though it is still a set of solutions.
Part 2: Reasoning and Context (Problems 4-6)
The point (a, 13) is on the graph of y = 4x - 3, and the point (-2, b) is on the graph of y = 3ˣ. Find a and b.
A bake sale sells muffins for $4 and pies for $5 and raises exactly $60, so 4m + 5p = 60. List every solution that makes sense for this sale, and explain why the graph in this context is a set of points instead of a full line.
Sam says (3, 4) is not on the graph of 3x - 2y = 1 because 3(4) - 2(3) = 6. Explain Sam's mistake, decide whether (3, 4) is on the graph, and find two more points that are on the graph.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Substitution
Coordinates substituted into the correct variables every time
One swap or arithmetic slip
No substitution shown
Conclusion
Correct on/off decision with a reason
Correct decision without a reason
Incorrect decision
Graphing
Accurate table and plot, curve connected where appropriate
Minor plotting errors
No graph
Explanation
Clearly links points on the graph to solutions of the equation
Explanation vague or incomplete
Missing
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
Which point is on the graph of y = 3x - 5?
Answer: A
Substitute x = 2: y = 3(2) - 5 = 1, so (2, 1) is a solution and is on the graph. Choice B swaps the coordinates: x = 1 gives y = -2, not 2. Choice C has the wrong sign: x = 0 gives y = -5. Choice D: x = 3 gives y = 4.
Question 2 of 20 · Multiple Choice
The point (4, k) is on the graph of 2x + y = 11. What is k?
Answer: C
Substitute x = 4 and y = k: 2(4) + k = 11, so k = 3. Choice A subtracts 4 instead of 2(4) = 8. Choice D treats 4 as the y-value: 2x + 4 = 11 gives x = 3.5.
Question 3 of 20 · Multiple Choice
What is the graph of an equation in two variables?
Answer: B
The graph is every solution (x, y) plotted in the coordinate plane. Choice A describes only the x-intercepts, which are just some of the solutions. Choice D is false: y = x² and x² + y² = 25 have curved graphs.
Question 4 of 20 · Multiple Choice
Which point is NOT on the graph of x² + y² = 50?
Answer: D
For (6, 4): 36 + 16 = 52, which is not 50. The others all work: 1 + 49 = 50, 25 + 25 = 50 and 49 + 1 = 50. Students who reject B or C may be squaring negatives incorrectly: (-5)² = 25 and (-1)² = 1.
Question 5 of 20 · Multiple Choice
The point (3, 8) is not on the graph of y = x² - 2. What does this mean?
Answer: B
Being off the graph means the pair is not a solution: 3² - 2 = 7, not 8. Choice C is false because (3, 7) is a solution. Choice D is false: 8² - 2 = 62, so (8, 3) is not on the graph either.
Question 6 of 20 · Multiple Choice
The point (-2, 9) lies on the graph of which equation?
Answer: A
(-2)² + 5 = 4 + 5 = 9, so the point satisfies y = x² + 5. Choice B gives -2(-2) + 3 = 7. Choice C gives (-2)³ + 1 = -7; a common error is treating (-2)³ as positive. Choice D gives -6 + 12 = 6.
Question 7 of 20 · Multiple Choice
A student made a table for y = 2ˣ. Which ordered pair from the table is NOT on the graph?
Answer: C
2³ = 8, so the correct point is (3, 8). The student computed 2 × 3 = 6 instead of 2³. The other pairs are correct: 2⁻¹ = 0.5, 2⁰ = 1 and 2² = 4.
Question 8 of 20 · Multiple Choice
Pens cost $2 and notebooks cost $5. The equation 2p + 5n = 30 describes spending exactly $30. Which pair (p, n) is a solution that also makes sense in this context?
Answer: A
2(5) + 5(4) = 10 + 20 = 30, and 5 pens with 4 notebooks is a real purchase. Choice C is a solution of the equation (15 + 15 = 30) but half a pen cannot be bought. Choice B swaps the numbers and gives 33. Choice D gives 25.
Question 9 of 20 · Multiple Choice
Which point is on the graph of y = |x - 1|?
Answer: D
|-2 - 1| = |-3| = 3, so (-2, 3) is on the graph. Choices A, B and C all have negative y-values, but an absolute value is never negative; those students forgot to take the absolute value (for example, 0 - 1 = -1).
Question 10 of 20 · Multiple Choice
How many solutions does the equation y = 2x + 1 have?
Answer: B
Every real x gives a matching y, so there are infinitely many ordered pairs, and together they form a line. Choice A confuses this with a one-variable equation such as 2x + 1 = 7, which has one solution.
Question 11 of 20 · Multiple Choice
The point (a, a), whose coordinates are equal, is on the graph of y = 3x - 8. What is a?
Answer: D
Substitute x = a and y = a: a = 3a - 8, so -2a = -8 and a = 4. Check: 3(4) - 8 = 4. Choice A comes from a sign error, dividing -8 by 2 instead of by -2.
Question 12 of 20 · Multiple Choice
Which equation has a graph that passes through both (0, -4) and (2, 0)?
Answer: C
For y = x² - 4: x = 0 gives -4 and x = 2 gives 0, so both points are solutions. Choice D has the opposite signs: it passes through (0, 4) and (2, 0). Choice B passes through (2, 0) but gives y = 4 at x = 0. Choice A gives -2 at x = 0.
Question 13 of 20 · Multiple Choice
Which point is on the graph of y = 1/x?
Answer: C
1/(-4) = -0.25, so (-4, -0.25) is a solution. Choice A fails because 1/0 is undefined, so no point on the graph has x = 0. Choice D is wrong because 1/0.5 = 2, not 0.2. Choice B gives 1/2, not 2.
Question 14 of 20 · Multiple Choice
In the coordinate plane, what is the graph of the equation x = 3?
Answer: B
As an equation in two variables, x = 3 is true for (3, y) with any y, such as (3, -2) and (3, 10). All those points form a vertical line. Choice A lists only one of the infinitely many solutions. Choice C is the graph of y = 3.
Question 15 of 20 · Short Answer
Is (-3, 2) on the graph of 4x + 5y = -2? Show your work.
Yes. 4(-3) + 5(2) = -12 + 10 = -2, which equals the right side, so (-3, 2) is a solution and lies on the line.
Question 16 of 20 · Short Answer
Find the points on the graph of y = -x² + 3 with x = -1, 0 and 2.
(-1, 2), (0, 3) and (2, -1) -(-1)² + 3 = -1 + 3 = 2, -(0)² + 3 = 3, and -(2)² + 3 = -4 + 3 = -1. A common error is computing -(-1)² as +1.
Question 17 of 20 · Short Answer
Explain why the graph of y = x² is drawn as a smooth curve instead of just the points (-2, 4), (-1, 1), (0, 0), (1, 1) and (2, 4).
The graph is the set of all solutions, and every real x-value, not just whole numbers, gives a solution. For example, (1.5, 2.25) and (-0.5, 0.25) are also solutions. Infinitely many solutions fill in the gaps between the table points, so they form an unbroken curve.
Question 18 of 20 · Short Answer
Find all points on the graph of x² + y² = 100 that have y-coordinate 6.
(8, 6) and (-8, 6) Substitute y = 6: x² + 36 = 100, so x² = 64 and x = 8 or x = -8. Two points on the circle share the same y-value.
Question 19 of 20 · Short Answer
A parking garage charges $3 plus $5 per hour, so the cost is C = 5h + 3. The point (4, 23) is on the graph. What does it mean? Is (6, 30) on the graph?
(4, 23) means parking for 4 hours costs $23: 5(4) + 3 = 23. (6, 30) is not on the graph, because 5(6) + 3 = 33. Six hours costs $33, so the point would be (6, 33).
Question 20 of 20 · Short Answer
Write an equation in two variables whose graph is not a line, and give three points on its graph. Show that each point is a solution.
Answers vary. Sample: y = x³ with points (-1, -1), (0, 0) and (2, 8), since (-1)³ = -1, 0³ = 0 and 2³ = 8. Any equation is acceptable if the three points are checked correctly by substitution and the graph is not a line (for example, a parabola, a circle or an absolute value graph).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
Is the graph of an equation the same as its set of solutions?
Yes: a graph and an equation carry the same information. Every point on the graph of an equation in x and y is a solution of the equation, and every solution is a point on the graph. The standard also notes that these solutions usually form a curve, and a line is one kind of curve.
Isn't this just "graphing lines" again?
No. Graphing lines is a procedure; this standard is about the meaning behind every graph students will draw. It covers any equation in two variables, including y = x² - 1, y = |x|, y = 1/x and x² + y² = 25. The test is always the same: substitute the coordinates and see whether the equation is true.
How do I check whether a point is on a graph?
Substitute the x-coordinate for x and the y-coordinate for y, then simplify both sides. If the two sides are equal, the point is on the graph. If not, it is not, however close it looks on a sketch. For example, (4, 1) looks close to the line 2x + 3y = 12, but 8 + 3 = 11, so it is not on it.
What is a common mistake?
Swapping the coordinates, that is, substituting the y-value for x. Remind students that an ordered pair is always (x, y). Another common mistake is deciding from a hand-drawn sketch instead of substituting, since a sketch cannot tell 2 from 2.1.
Why do we connect the dots when we graph from a table?
Because the table shows only a few of the solutions. When every real number between the table values also gives a solution, those extra points fill in the gaps and the graph is an unbroken curve. When some x-values give no solution, such as x = 0 for y = 1/x, the graph has a break there.
Can the graph of an equation fail to be a function?
Yes. The graph of x² + y² = 25 is a circle, and the vertical line x = 3 crosses it at (3, 4) and (3, -4). It is still exactly the set of solutions of its equation. This is a good way to separate "graph of an equation" from "graph of a function" in students' minds.
In a word problem, why don't we use every point on the line?
The equation's graph includes every real solution, but a context may allow only some of them. For 2g + 3j = 24 with granola bars and juice boxes, (1.5, 7) is a solution of the equation, but you cannot buy half a bar. The sensible solutions are the whole-number points (0, 8), (3, 6), (6, 4), (9, 2) and (12, 0).
How does HSA.REI.D.10 connect to solving systems?
If a point is on two graphs, it satisfies both equations, so it solves the system. That is why intersection points solve systems of equations (HSA.REI.C.6 and HSA.REI.C.7) and why the x-coordinates of intersections of y = f(x) and y = g(x) solve f(x) = g(x) (HSA.REI.D.11).
How is HSA.REI.D.10 tested?
Typical items give a point and ask whether it is on a graph, give a graph and ask which equation it could represent, or give a point on a graph with one missing coordinate. On the digital SAT, questions in the Algebra and Advanced Math domains often expect students to use the fact that a point on a graph satisfies its equation, for example to find an unknown constant.
What can parents do to help with this topic?
Ask your student to explain a graph from homework in words: "Pick a point on this curve. What do its numbers mean, and why do they make the equation true?" If they can substitute and explain the result, they understand the standard.
07
Related Standards
6 standards
These standards connect to HSA.REI.D.10: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.A.1Prerequisite
A function assigns one output to each input; its graph is the set of input-output pairs
