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

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

An employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled 1 large basket, using a total of 18 pieces of fruit. Using wire baskets, the employee assembled 7 small baskets and 9 large baskets, using a total of 204 pieces of fruit. Assuming that each small basket includes the same amount of fruit, as does every large basket, how many pieces are in each?

The small baskets each include pieces and the large ones each include pieces.

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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. Steve selected 1 box of ballpoint pens, paying $2. Next, Tara spent $15 on 9 boxes of pencils and 3 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.

y = 2

9x + 3y = 15

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.

0x + 1y = 2

9x + 3y = 15

→

0	1		2
9	3		15

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.

0	1		2
9	3		15

9	3		15
0	1		2

Swap R₁ with R₂

Multiply R₁ by

1	0		1
0	1		2

Add

^–1

R₂

to R₁

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		1
0	1		2

The solution is (1, 2).

A box of pencils costs $1, and a box of pens costs $2.

Looking for more? Try this:

$Confetti$

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N.8Solve a system of equations using augmented matrices

N.7Solve a system of equations using elimination: word problems

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