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

Solve using augmented matrices.

10x − 9y = ^–1

x = 8

(, )

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question

Solve using augmented matrices.

^–3x + 3y = 18

x = ^–9

(, )

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

Use augmented matrices to solve the system of equations:

^–3x + 3y = 18

x = ^–9

Step 1: Write the augmented matrix.

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

^–3x + 3y = 18

1x + 0y = ^–9

→

^–3	3		18
1	0		^–9

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.

^–3	3		18
1	0		^–9

1	0		^–9
^–3	3		18

Swap R₁ with R₂

1	0		^–9
0	3		^–9

Add 3R₁ to R₂

1	0		^–9
0	1		^–3

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		^–9
0	1		^–3

The solution is (^–9, ^–3).

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