IXL - Solve a system of equations using augmented matrices: word problems (Algebra 1 practice)

Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

The administrative assistant at a software company often provides breakfast when there is a morning meeting. For last week's sales meeting, she purchased 8 dozen doughnuts and 4 dozen croissants, spending a total of $80. In preparation for yesterday's safety meeting, she spent $10 on 1 dozen croissants. Assuming she purchased the items at the same bakery both times, how much does a dozen of each cost?

A dozen doughnuts costs $, and a dozen croissants costs $.

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Learn with an example

question

Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

Two coworkers picked up some writing instruments at the office supply store. Rodrigo selected 1 box of pencils, paying $2. Next, Audrey spent $14 on 3 boxes of pencils and 4 boxes of ballpoint pens. How much does a box of each cost?

A box of pencils costs $, and a box of pens costs $.

key idea

To solve using augmented matrices, follow these three steps:

Step 1: Write the augmented matrix.

Step 2: Use elementary row operations to transform the left part of the augmented matrix into the identity matrix.

Step 3: State the solution.

solution

Before you can solve, you must write a system of equations. Let x represent the cost of a box of pencils, and let y represent the cost of a box of pens.

x = 2

3x + 4y = 14

Now use augmented matrices to solve the system of equations.

Step 1: Write the augmented matrix.

Since both equations are in standard form, you can write the numbers in an augmented matrix.

1x + 0y = 2

3x + 4y = 14

→

1	0		2
3	4		14

Step 2: Use elementary row operations to transform the left part of the augmented matrix into the identity matrix.

There are three elementary row operations:

Swap any two rows

Multiply any row by a non-zero number

Add a multiple of any row to any other row

Use these three operations to make the part of the matrix to the left of the dashed line look like the identity matrix:

1	0
0	1

Start with the augmented matrix from Step 1.

1	0		2
3	4		14

1	0		2
0	4		8

Add ^–3R₁ to R₂

1	0		2
0	1		2

Multiply R₂ by

The left part is now the identity matrix, so no more row operations need to be performed.

Step 3: State the solution.

The solution is the last column of the new matrix.

1	0		2
0	1		2

The solution is (2, 2).

A box of pencils costs $2, and a box of pens costs $2.

Looking for more? Try this:

$Confetti$

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Work it out

Not feeling ready yet? These can help:

O.12Solve a system of equations using augmented matrices

N.8Write a linear function: word problems

C.9Solve one-step and two-step linear equations: word problems

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