Solve 2x + 3y = 12, x + 2y = 7 using the inverse matrix Step by Step

Step by step

The question

Question: solve 2x + 3y = 12 and x + 2y = 7 using the inverse matrix.

2x + 3y = 12

We want the x and y that make both true.

x + 2y = 7

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.

[2, 3; 1, 2] × [x; y] = [12; 7]

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 = |2, 3; 1, 2|

So D = 2 × 2 − 1 × 3.

D = 2 × 2 − 1 × 3

2 times 2 is 4.

1 times 3 is 3.

D = 4 − 3

4 minus 3 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 × [2, −3; −1, 2]

So A⁻¹ = rows 2, −3 and −1, 2.

A⁻¹ = [2, −3; −1, 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [2, −3; −1, 2] × [12; 7]

= [2 × 12 + (−3) × 7; −1 × 12 + 2 × 7]

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

[x; y] = [3; 2]

So the answer is (3, 2).

(x, y) = (3, 2)

4 Check

Let's check in every equation.

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

Equation 1 2(3) + 3(2) = 6 + 6 = 12

We get 12, the same as the right side.

Now equation 2.

Equation 2 3 + 2(2) = 3 + 4 = 7

We get 7, 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