HSA.REI.C.8: Writing a System of Linear Equations as a Matrix Equation
In plain English: HSA.REI.C.8 is an advanced (+) Common Core algebra standard that asks students to write a system of linear equations as one matrix equation AX = B, where A holds the coefficients, X is a column vector of the variables and B is a column vector of the constants. It is usually taught in Precalculus or an honors Algebra II course, just before solving systems with inverse matrices.
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
Common Core State Standards for Mathematics · Domain: Reasoning with Equations and Inequalities (REI) · Cluster: Solve systems of equations Also written as HSA-REI.C.8 or A-REI.8 · Official standard
Students learn to represent a system of linear equations as a single matrix equation AX = B. The coefficient matrix A has one row per equation and one column per variable, X is the column vector of variables, and B is the column vector of constants. Students first put each equation in standard form, with the variables in the same order on the left side and the constant on the right, and they write a 0 for any variable that is missing from an equation.
The key idea is that multiplying the matrix A by the vector X row by row gives back the left sides of the equations, so the one matrix equation says exactly what the whole system says. Students move in both directions, from a system to a matrix equation and from a matrix equation back to a system, work with two and three variables, model context problems, and check a proposed solution vector by computing AX. Matrices are written row by row, with rows separated by semicolons: [5 2; 3 -4] has first row 5, 2 and second row 3, -4, and the column vector [x; y] has x on top of y.
Learning Objectives
By the end of this lesson, students will be able to:
Rewrite each equation of a linear system in standard form with the variables in a fixed order, using 0 for a missing variable
Write a system of two or three linear equations as a single matrix equation AX = B and name the coefficient matrix, the variable vector and the constant vector
Translate a matrix equation back into the system of equations it represents
Explain why AX = B represents the system by multiplying A by X row by row, and use that product to test whether a vector is a solution
Model a context problem with a system and write it as a matrix equation with clearly defined variables
Prior Knowledge Required
Students should already be comfortable with:
Writing and solving systems of two linear equations in two variables 8.EE.C.8
Solving systems of linear equations algebraically HSA.REI.C.6
Multiplying a matrix by a column vector HSN.VM.C.11
Matrix dimensions (rows × columns) and naming entries by row and column
Post two short tasks. The first reviews the matrix-vector product from HSN.VM.C.11; the second sets up the idea of the lesson.
Warm-Up Prompt
"(1) Compute [2 1; 3 -1][4; 5]. (2) A coffee cart sells small coffees for $3 and large coffees for $4. On Monday it sold 50 coffees and took in $172. Write a system of equations. Then look at only these six numbers: 1, 1, 50, 3, 4, 172. Could someone rebuild your system from them? What else would they need to know?"
For (1), students should get [2(4) + 1(5); 3(4) - 1(5)] = [13; 7]. For (2), with s small and l large coffees: s + l = 50 and 3s + 4l = 172. Students usually notice that the numbers alone are not enough: they also need to know which number goes with which variable and which number is the total. That bookkeeping is exactly what a matrix equation records: the position of each number tells you its role.
Direct Instruction15 minutes
Use Diagram 1 to name the three parts of a matrix equation, then give the procedure:
Put every equation in standard form: all variable terms on the left in the same order (for example x, y, z), the constant on the right.
Fill in hidden coefficients: a variable written alone has coefficient 1 (or -1 if it is subtracted), and a missing variable has coefficient 0.
Build the coefficient matrix A: row i holds the coefficients of equation i; column j holds the coefficients of variable j. A system of m equations in n variables gives an m × n matrix.
Write the variable vector X and the constant vector B: X lists the variables in the same order as the columns of A; B lists the right sides in the same order as the rows.
Write AX = B and check it: multiply A by X row by row. Each row of the product must be the left side of the matching equation.
Work through the examples below. For each one, ask students to predict the dimensions of A before writing it.
Two equations, already in standard form
Write 5x + 2y = 16 and 3x - 4y = 20 as a matrix equation.
Equation: [5 2; 3 -4][x; y] = [16; 20]
Rearrange first
Write y = 3x - 7 and 2y + x = 14 as a matrix equation. Standard form: -3x + y = -7 and x + 2y = 14.
Equation: [-3 1; 1 2][x; y] = [-7; 14]
Three variables with missing terms
Write x + 2y - z = 4, 3y + 2z = 1 and 4x - z = 9 as a matrix equation. The first equation has no missing variable; the second has no x; the third has no y.
A school play sold 240 tickets: adult tickets a at $12 and student tickets s at $7, for a total of $2,280. Write a matrix equation with X = [a; s].
Equation: [1 1; 12 7][a; s] = [240; 2280]
Matrix equation back to a system
Write the system represented by [2 0 -3; 1 -1 0; 0 4 1][p; q; r] = [5; -2; 8].
Equation: 2p - 3r = 5, p - q = -2, 4q + r = 8
After the first example, use Diagram 2 to multiply A by X row by row: the first row of AX is 5x + 2y and the second is 3x - 4y, so AX = B says the same thing as the two equations. Substituting the vector [4; -2] for X shows that it satisfies both rows, so (4, -2) is the solution of the system. In the context example, point out that the variable vector fixes the meaning of each column: column 1 is about adult tickets and column 2 about student tickets.
Guided Practice15 minutes
Pairs write each system as a matrix equation, one at a time, and state the dimensions of A. (a) 4x - y = 9 and x + 3y = -1 (answer: [4 -1; 1 3][x; y] = [9; -1]). (b) 2x = 5y + 3 and y - x = 4 (standard form 2x - 5y = 3 and -x + y = 4, so [2 -5; -1 1][x; y] = [3; 4]). (c) x + y + z = 6, 2x - z = 1 and y + 3z = 11 (answer: [1 1 1; 2 0 -1; 0 1 3][x; y; z] = [6; 1; 11]). After each one, a pair multiplies A by X at the board to confirm the rows. Listen for these errors: copying a coefficient without its sign, reading the numbers in the order they appear instead of by variable, leaving a gap instead of writing 0, and putting the constants into A.
Independent Practice15 minutes
Students work alone on five items: (1) 9x + 4y = 1 and 2x - 7y = 12 (answer: [9 4; 2 -7][x; y] = [1; 12]); (2) x = 2y - 6 and 3y = x + 10 (standard form x - 2y = -6 and -x + 3y = 10, so [1 -2; -1 3][x; y] = [-6; 10]); (3) 2x - y + z = 0, x + 3z = 8 and 5y - 2z = -1 (answer: [2 -1 1; 1 0 3; 0 5 -2][x; y; z] = [0; 8; -1]); (4) write the system for [1 -3; 4 2][u; v] = [7; 0] (answer: u - 3v = 7 and 4u + 2v = 0); (5) a gym sold 90 memberships and day passes in June, memberships at $35 and day passes at $8, for $2,070 in total. Write the matrix equation with X = [m; d] (answer: [1 1; 35 8][m; d] = [90; 2070]). Early finishers show that [50; 40] satisfies the equation in (5) by computing AX.
Closure5 minutes
Exit ticket: (1) Write 7x - 2y = 3 and y = 4x + 1 as a matrix equation. (Answer: [7 -2; -4 1][x; y] = [3; 1].) (2) Write the system represented by [3 -1; 0 2][u; v] = [8; -6]. (Answer: 3u - v = 8 and 2v = -6.) (3) In one sentence, explain why the order of the variables in X must match the order of the columns of A.
Differentiation Strategies
For Struggling Students
Give a template with labeled column headings (x, y, z, =) above empty boxes, so students write each equation into a row before adding brackets
Have students circle each coefficient together with its sign before copying it into A
Start with systems already in standard form with no missing variables, then add one rearrangement step at a time
For Advanced Students
Ask students to write a system of 2 equations in 3 variables as a matrix equation and explain why A is not square
Ask how the matrix equation changes if the variables are listed as [y; x] instead of [x; y], and why both versions describe the same system
Preview HSA.REI.C.9 as a challenge: if A has an inverse, what single matrix product would give X?
Assessment Guidance
What to Look For
Check that students rewrite equations in standard form before building A, that every row of A has one entry per variable (including zeros), and that the signs of coefficients are kept. Ask students to multiply A by X to justify their matrix equation: a student who can explain that each row of AX is the left side of one equation understands the representation, not only the layout. In context problems, look for a variable vector with defined variables, such as X = [a; s] where a is the number of adult tickets.
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Classroom Activities
3 Activities
1
Color-Code the Coefficients
15 minPairs
Students use three colors to separate coefficients, variables and constants in four systems, then copy each color into its own part of the matrix equation. The colors make the structure of AX = B visible before students work without them.
Procedure
Give each pair four systems: (i) 6x + 5y = 2 and x - y = 9; (ii) 3y = 12 - 2x and x = y + 1; (iii) 2a + 3b - c = 5, a - 4c = 0 and b + c = 7; (iv) x/2 + y = 3 and x - y/3 = 4
Partner A rewrites each system in standard form; Partner B checks it and circles coefficients in blue, variables in black and constants in a third color
Together they write A (blue), X (black) and B (third color). Expected answers: (i) [6 5; 1 -1][x; y] = [2; 9]; (ii) [2 3; 1 -1][x; y] = [12; 1]; (iii) [2 3 -1; 1 0 -4; 0 1 1][a; b; c] = [5; 0; 7]; (iv) [1/2 1; 1 -1/3][x; y] = [3; 4]
Each pair multiplies A by X for system (iii) and shows that the rows match the three equations
Discussion Questions
In system (iii), where did the zeros in A come from?
In system (iv), the coefficients are fractions. Is A still a coefficient matrix? Could you clear the fractions first, and how would A and B change?
What would go wrong if you wrote the constants of system (ii) before rearranging?
Modification for Distance Learning
Share a slide with the four systems and let students use highlighter colors in a shared document. Pairs paste their finished matrix equations into a class slide for comparison.
2
Matrix Equation Card Sort
20 minGroups of 3-4
Each group gets 12 cards: 4 systems as they were first written, 4 of the same systems in standard form, and 4 matrix equations. Groups build 4 matching triples and justify each match by multiplying A by X.
Set 2: "y = 2x + 5, 4x + y = -1" / "-2x + y = 5, 4x + y = -1" / [-2 1; 4 1][x; y] = [5; -1]
Set 3: "2x = z + 3, x + y = 4, 5z = -y" / "2x - z = 3, x + y = 4, y + 5z = 0" / [2 0 -1; 1 1 0; 0 1 5][x; y; z] = [3; 4; 0]
Set 4: "5 - x = 2y, y - 3 = x" / "x + 2y = 5, -x + y = 3" / [1 2; -1 1][x; y] = [5; 3]
Procedure
Shuffle the cards and deal them out; students take turns placing a card next to one it matches and explaining why
A match counts only after another group member multiplies A by X and reads the rows aloud
When all 4 triples are built, each group writes one new "first written" card for a classmate group to sort
Challenge Variation
Add two decoy matrix cards, [1 3; 4 -2][x; y] = [10; 2] (columns of Set 1 mixed up) and [2 -1 0; 1 1 0; 0 1 5][x; y; z] = [3; 4; 0] (the -1 of Set 3 placed in the y column). Groups must explain which system each decoy actually represents.
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Model It, Then Test a Solution
20 minGroups of 3
Groups turn two context problems into matrix equations, then use a calculator to multiply A by a proposed solution vector and decide whether it is correct. This connects the representation to its purpose: the whole system is now one object that a calculator can work with.
Scenarios
A school store sold 70 items: T-shirts at $12, water bottles at $8 and pennants at $5, for $540 in total. It sold twice as many pennants as T-shirts. Define t, w and p, then write the matrix equation. (Answer: [1 1 1; 12 8 5; -2 0 1][t; w; p] = [70; 540; 0].) Test the vector [10; 40; 20]
A band concert sold 400 floor tickets at $25 and bleacher tickets at $15, for $7,600 in total. (Answer: with f floor and b bleacher tickets, [1 1; 25 15][f; b] = [400; 7600].) Test the vectors [160; 240] and [150; 250]
Procedure
Roles: the modeler defines the variables and writes the equations, the builder writes A, X and B, and the tester enters A and the proposed vector on a calculator and computes the product
The group compares the product with B entry by entry and writes one sentence about what each matching or non-matching entry means in the context
Groups then write their own three-variable context problem and trade with another group
Discussion Questions
For the vector [150; 250], which entry of AX matched B and which did not? What does that mean about the tickets and the money?
Why does the pennant condition "twice as many pennants as T-shirts" give a row with a 0 on the right side?
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Diagrams & Visual Aids
2 diagrams
Diagram 1: The Three Parts of a Matrix Equation
The system 5x + 2y = 16 and 3x - 4y = 20 becomes AX = B. The coefficient matrix A has one row per equation and one column per variable, the variable vector X lists the variables in column order, and the constant vector B lists the right sides in row order.
Diagram 2: Why AX = B Says the Same Thing as the System
Multiplying A by X row by row gives the left side of each equation, so AX = B holds exactly when both equations hold. Replacing X by the vector [4; -2] gives [16; 20] = B, so (4, -2) is the solution of the system. The green note confirms that every row matches.
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Homework Assignment
~30 min
HSA.REI.C.8 Homework: Systems as Matrix Equations
Directions: Show your standard-form equations before writing any matrix. For every matrix equation, label A, X and B and state the dimensions of A. Write matrices row by row with rows separated by semicolons, for example [5 2; 3 -4]. A calculator may be used for Problems 4 and 5.
Part 1: From a System to a Matrix Equation (Problems 1-3)
Write each system as a matrix equation AX = B. (a) 6x + y = 11 and 2x - 5y = -7 (b) 3m - 2n = 0 and m + n = 15, with X = [m; n].
Rewrite each system in standard form, then write it as a matrix equation. (a) y = -2x + 9 and 3x = y + 1 (b) 4y - 3 = 2x and x + 2 = 5y
Write the system x - y + 2z = 3, 2x + z = 7 and y - 4z = -10 as a matrix equation. Explain where each 0 in A comes from.
Part 2: Reading, Testing and Modeling (Problems 4-6)
Write the system of equations represented by [4 -1 0; 2 0 3; 0 1 -5][x; y; z] = [6; 13; -9]. Then decide whether (x, y, z) = (2, 2, 3) is a solution by computing A times the vector [2; 2; 3]. If it is not, say which equation fails.
A theater sold 600 tickets: orchestra seats at $40, mezzanine seats at $30 and balcony seats at $20, for $19,000 in total. It sold twice as many orchestra seats as balcony seats. Let x, y and z be the numbers of orchestra, mezzanine and balcony tickets. Write the system and the matrix equation, then show that [200; 300; 100] satisfies it.
A student writes the system 2x + 3y = 8 and y - 4x = 1 as [2 3; 1 -4][x; y] = [8; 1]. Explain the error, write the correct matrix equation, and write the system that the student's matrix equation actually represents.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Standard Form
Every equation rewritten correctly with variables in a fixed order
One rearrangement or sign error
Equations not rewritten
Matrix Equation
A, X and B correct and labeled, dimensions stated, zeros included
One entry wrong or a label missing
Structure of AX = B missing
Reading and Testing
System read back correctly and AX computed to test the vector
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. Matrices are written row by row, with rows separated by semicolons: [5 2; 3 -4] has first row 5, 2 and second row 3, -4, and the column vector [x; y] has x on top of y.
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
What is the coefficient matrix of the system 7x - 3y = 2 and x + 6y = 11?
Answer: B
Row 1 holds the coefficients of the first equation (7 and -3), and row 2 holds those of the second (1 and 6). Choice A drops the minus sign on -3y. Choice C is the augmented matrix: it includes the constants, which belong in B. Choice D writes the equations as columns instead of rows.
Question 2 of 20 · Multiple Choice
What is the constant vector B for the system 2x + 9y = -4 and 5x - y = 13?
Answer: D
B is a column vector of the right sides, in the same order as the equations: [-4; 13]. Choice A is the first column of A (the coefficients of x), and choice B is the second column. Choice C has the right numbers but is a row, so its dimensions (1 × 2) do not match AX, which is 2 × 1.
Question 3 of 20 · Multiple Choice
A system of three equations uses the variables p, q and r in that order. What is the variable vector X?
Answer: A
X is a 3 × 1 column vector that lists the variables in the order of the columns of A. Choice B is a row vector, so the product AX is not defined for a 3 × 3 matrix A. Choice C lists coefficients, not variables. Choice D leaves out r.
Question 4 of 20 · Multiple Choice
Which matrix equation represents the system y = 5x - 2 and x + y = 10?
Answer: C
Rewrite the first equation in standard form: -5x + y = -2. With X = [x; y], row 1 of A is [-5 1] and row 2 is [1 1], and B = [-2; 10]. Choice A forgets to change the sign of 5x when moving it to the left side. Choice B lists the coefficients in the order y, x while X is still [x; y]. Choice D changes the sign of the constant as well as the x term.
Question 5 of 20 · Multiple Choice
Which system is represented by [3 0; -2 4][x; y] = [9; 2]?
Answer: B
Multiply row by row: row 1 gives 3x + 0y = 3x, so 3x = 9; row 2 gives -2x + 4y, so -2x + 4y = 2. Choice A reads the columns as equations. Choice C treats the second column as constants instead of coefficients of y. Choice D matches -2 with y and 4 with x, the reverse of the column order.
Question 6 of 20 · Multiple Choice
A system has 3 equations in 4 variables. What are the dimensions of its coefficient matrix A?
Answer: C
A has one row per equation and one column per variable, so it is 3 × 4, X is 4 × 1 and B is 3 × 1. Choice A swaps rows and columns. Choice D counts the column of constants, which gives the augmented matrix, not A.
Question 7 of 20 · Multiple Choice
What is the coefficient matrix of x + 4z = 7, 2x - y = 3 and y + z = 5, with X = [x; y; z]?
Answer: B
Each row needs an entry for x, y and z. The first equation has no y, the second has no z and the third has no x, so those entries are 0: [1 0 4; 2 -1 0; 0 1 1]. Choice A pushes the coefficients to the left instead of keeping them under their variables. Choice C leaves the zeros out, so the columns no longer belong to single variables. Choice D loses the minus sign of -y.
Question 8 of 20 · Multiple Choice
The equation 4x - 7 = 3y is one equation in a system with X = [x; y]. Which row of A and which entry of B come from it?
Answer: D
Subtract 3y and add 7 on both sides: 4x - 3y = 7. So the row of A is [4 -3] and the entry of B is 7. Choice B moves 3y but leaves the constant with its old sign. Choice A forgets that 3y changes sign when it moves to the left. Choice C treats -7 as a coefficient.
Question 9 of 20 · Multiple Choice
Is (x, y) = (3, -1) a solution of [2 5; 1 -3][x; y] = [1; 6]?
Answer: A
Row 1: 2(3) + 5(-1) = 1. Row 2: 1(3) - 3(-1) = 6. The product A[3; -1] = [1; 6] = B, so the vector is a solution. Choice B adds 5 instead of multiplying by -1. Choice D misses the point of the representation: testing a vector only takes one matrix-vector product.
Question 10 of 20 · Multiple Choice
Pens cost $2 and notebooks cost $5. A customer buys 12 items for $39. With X = [p; n] for pens and notebooks, which matrix equation models the purchase?
Answer: D
The equations are p + n = 12 (items) and 2p + 5n = 39 (dollars), so A = [1 1; 2 5] and B = [12; 39]. Choice A writes the equations as columns. Choice B swaps the prices, which would mean pens cost $5. Choice C swaps the entries of B, so each row no longer matches its total.
Question 11 of 20 · Multiple Choice
In a matrix equation AX = B with X = [x; y; z], what does the entry in row 2, column 1 of A tell you?
Answer: A
Rows of A correspond to equations and columns to variables, so row 2, column 1 is the coefficient of the first variable, x, in the second equation. Choice B reads the position as column 2, row 1. Choice C describes the second entry of B. Choice D confuses the coefficient matrix with the solution vector.
Question 12 of 20 · Multiple Choice
The system 3x + 2y = 8 and x - y = 1 is written with the variable vector X = [y; x]. Which coefficient matrix goes with this X?
Answer: B
Column 1 must hold the coefficients of y (2 and -1) and column 2 those of x (3 and 1), so A = [2 3; -1 1]. Choice A is the matrix for X = [x; y]; it is correct only with that order. Choice C moves only the first row. Choice D also swaps the rows, which would require swapping the entries of B.
Question 13 of 20 · Multiple Choice
For A = [1 1 1; 2 -1 0; 0 3 1] and X = [x; y; z], what is the product AX?
Answer: C
Multiply each row by the column: row 1 gives x + y + z, row 2 gives 2x - y + 0z, row 3 gives 0x + 3y + z. Choice A multiplies the columns of A by X instead of the rows. Choice B adds the entries in each row and attaches only one variable. Choice D treats the 0 in row 2 as a 1.
Question 14 of 20 · Multiple Choice
Why must every equation be in standard form, with the variables in the same order, before you write A?
Answer: D
The product AX multiplies column j of A by the j-th variable, so every entry in a column must belong to the same variable, and the constants must be on the right side to form B. Choice A is false: a system of 3 equations in 2 variables has a 3 × 2 matrix. Choice B is false: negative entries are common. Choice C is unrelated to the form of the equations.
Question 15 of 20 · Short Answer
Write the system 8x - 3y = 5 and 2x + y = -4 as a matrix equation. Name A, X and B.
[8 -3; 2 1][x; y] = [5; -4], with A = [8 -3; 2 1], X = [x; y] and B = [5; -4]. Check: A times X is [8x - 3y; 2x + y], the left sides of the two equations.
Question 16 of 20 · Short Answer
Write the system of equations represented by [1 -2 3; 0 5 -1; 4 0 2][x; y; z] = [0; 7; -6].
Multiply row by row: x - 2y + 3z = 0, 5y - z = 7 and 4x + 2z = -6. The zeros mean that the second equation has no x term and the third has no y term.
Question 17 of 20 · Short Answer
Rewrite 3y = 2x + 12 and x = 4 - y in standard form, then write the system as a matrix equation with X = [x; y].
Standard form: -2x + 3y = 12 and x + y = 4. Matrix equation: [-2 3; 1 1][x; y] = [12; 4]. Multiplying A by X gives [-2x + 3y; x + y], which matches the left sides.
Question 18 of 20 · Short Answer
A cafe sells small drinks for $2, medium for $3 and large for $4. In one hour it sold 45 drinks for $139, and it sold 5 more medium drinks than small drinks. Define variables and write a matrix equation for this situation.
Let s, m and l be the numbers of small, medium and large drinks. The equations are s + m + l = 45, 2s + 3m + 4l = 139 and m - s = 5, or -s + m = 5. Matrix equation: [1 1 1; 2 3 4; -1 1 0][s; m; l] = [45; 139; 5]. (The solution is 12 small, 17 medium and 16 large drinks, which students can confirm by computing A[12; 17; 16].)
Question 19 of 20 · Short Answer
A student writes x + y = 6 and 2x = 10 as [1 1; 2][x; y] = [6; 10]. What is wrong, and what is the correct matrix equation?
The second row has only one entry, so A is not a matrix and the product AX is not defined. The second equation has no y, so its y-coefficient is 0: 2x + 0y = 10. Correct equation: [1 1; 2 0][x; y] = [6; 10].
Question 20 of 20 · Short Answer
Show by multiplication that X = [2; 4] is a solution of [2 -1; 3 4]X = [0; 22]. What system does this matrix equation represent?
Row 1: 2(2) - 1(4) = 0. Row 2: 3(2) + 4(4) = 6 + 16 = 22. So AX = [0; 22] = B, and (2, 4) is a solution. The matrix equation represents the system 2x - y = 0 and 3x + 4y = 22.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does HSA.REI.C.8 mean?
HSA.REI.C.8 means students can rewrite a whole system of linear equations as one matrix equation, AX = B. A is the matrix of coefficients, X is a column vector of the variables, and B is a column vector of the constants. For example, 5x + 2y = 16 and 3x - 4y = 20 become [5 2; 3 -4][x; y] = [16; 20]. The standard is about the representation itself; solving the matrix equation with an inverse matrix is the next standard, HSA.REI.C.9.
Is HSA.REI.C.8 taught in Algebra 2 or Precalculus?
It is usually taught in Precalculus or in an honors Algebra II course, together with the matrix standards HSN.VM.C.6 to HSN.VM.C.12. It is marked (+), which Common Core uses for additional mathematics that students need for advanced courses such as calculus or discrete mathematics. Some integrated course sequences place it in Math 3 or a fourth-year course.
What is a vector variable?
A vector variable is one symbol, usually X or x with an arrow, that stands for the whole list of unknowns written as a column, such as X = [x; y; z]. Writing the unknowns as one vector is what lets a system of several equations become a single equation AX = B, in the same way that a single number variable turns one relationship into one equation.
Why write a system as a matrix equation at all?
A matrix equation packs the whole system into one object. That makes it possible to solve the system with one operation, X = A-1B, when A has an inverse (HSA.REI.C.9), and it is how calculators, spreadsheets and computer programs store and solve large systems. It also shows the structure of the system: the coefficients, the unknowns and the constants each have their own place.
What is the difference between the coefficient matrix and the augmented matrix?
The coefficient matrix A holds only the coefficients of the variables. The augmented matrix adds the constants as an extra column, for example [5 2 | 16; 3 -4 | 20]. The matrix equation AX = B uses the coefficient matrix, with the constants kept separately in B. The augmented matrix is used in row reduction, which is a different method.
What are common mistakes when writing AX = B?
A common error is building A before rewriting the equations in standard form, so a term such as 3y on the right side ends up with the wrong sign or in the wrong column. Other frequent errors are leaving out the 0 for a missing variable, dropping a minus sign, writing the equations as columns instead of rows, and writing X or B as a row instead of a column. Asking students to multiply A by X to check their work catches all of these.
Does the order of the variables matter?
Yes, the order must be consistent. You may list the variables in any order, but the columns of A must follow the order of X. For 4x + 5y = 9 and 2x - y = 3, both [4 5; 2 -1][x; y] = [9; 3] and [5 4; -1 2][y; x] = [9; 3] are correct. Mixing the orders, such as [5 4; -1 2][x; y], describes a different system.
Does the coefficient matrix have to be square?
No. A system of m equations in n variables gives an m × n coefficient matrix, so 2 equations in 3 variables give a 2 × 3 matrix. The matrix equation still represents the system. Only square matrices can have inverses, so the inverse method of HSA.REI.C.9 applies when the number of equations equals the number of variables.
How can students check that a matrix equation is right?
Multiply A by X row by row. Each row of the product must be the left side of one equation, and the matching entry of B must be its right side. Students can also test a known solution: if a vector satisfies every original equation, the product of A and that vector must equal B.
How does this standard connect to other courses?
It builds on solving systems in grade 8 and Algebra I (8.EE.C.8 and HSA.REI.C.6) and on matrix-vector multiplication (HSN.VM.C.11). It leads directly to solving systems with inverse matrices (HSA.REI.C.9), and later to linear algebra, where the same equation AX = B is studied for any number of equations and variables. Computer graphics, economics and engineering use this notation for large systems.
07
Related Standards
5 standards
These standards connect to HSA.REI.C.8: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.EE.C.8Prerequisite
Analyze and solve pairs of simultaneous linear equations
