Solve x + 2y = 8, x + 3y = 11 using the inverse matrix Step by Step

Step by step

The question

Question: solve x + 2y = 8 and x + 3y = 11 using the inverse matrix.

x + 2y = 8

We want the x and y that make both true.

x + 3y = 11

1 Write it as AX = B

The numbers in front of x and y make the matrix A.

The letters make X.

The right sides make B.

Row times column gives back the equations.

[1, 2; 1, 3] × [x; y] = [8; 11]

To get X alone, we multiply by the inverse of A: X = A⁻¹B.

AX = B → X = A⁻¹B

Careful!

The inverse goes in front of B.

With matrices, the order matters.

Careful: X = A⁻¹B, not BA⁻¹

2 Find the inverse

First D, the determinant of A.

D = |1, 2; 1, 3|

So D = 1 × 3 − 1 × 2.

D = 1 × 3 − 1 × 2

1 times 3 is 3.

1 times 2 is 2.

D = 3 − 2

3 minus 2 is 1.

So D = 1.

D = 1

Swap the top left and bottom right.

Change the signs of the other two.

Then times 1 over D.

A⁻¹ = 11 × [3, −2; −1, 1]

So A⁻¹ = rows 3, −2 and −1, 1.

A⁻¹ = [3, −2; −1, 1]

3 Multiply

X is A⁻¹ times B.

[x; y] = [3, −2; −1, 1] × [8; 11]

= [3 × 8 + (−2) × 11; −1 × 8 + 1 × 11]

Work them out: x = 2 and y = 3.

[x; y] = [2; 3]

So the answer is (2, 3).

(x, y) = (2, 3)

4 Check

Let's check in every equation.

We put x = 2, y = 3 into the first one.

Equation 1 2 + 2(3) = 2 + 6 = 8

We get 8, the same as the right side.

Now equation 2.

Equation 2 2 + 3(3) = 2 + 9 = 11

We get 11, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: write it as AX = B.

Then: find the inverse.

Then: multiply.

Last: check.

Answer