Solve 2x + 2y = 6, x + 3y = 7 using the inverse matrix Step by Step

Step by step

The question

Question: solve 2x + 2y = 6 and x + 3y = 7 using the inverse matrix.

2x + 2y = 6

We want the x and y that make both true.

x + 3y = 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, 2; 1, 3] × [x; y] = [6; 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, 2; 1, 3|

So D = 2 × 3 − 1 × 2.

D = 2 × 3 − 1 × 2

2 times 3 is 6.

1 times 2 is 2.

D = 6 − 2

6 minus 2 is 4.

So D = 4.

D = 4

Swap the top left and bottom right.

Change the signs of the other two.

Then times 1 over D.

A⁻¹ = 14 × [3, −2; −1, 2]

So A⁻¹ = rows 3/4, −1/2 and −1/4, 1/2.

A⁻¹ = [3 over 4, −1 over 2; −1 over 4, 1 over 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [3 over 4, −1 over 2; −1 over 4, 1 over 2] × [6; 7]

= [3 over 4 × 6 + (−1 over 2) × 7; −1 over 4 × 6 + 1 over 2 × 7]

Work them out: x = 1 and y = 2.

[x; y] = [1; 2]

So the answer is (1, 2).

(x, y) = (1, 2)

4 Check

Let's check in every equation.

We put x = 1, y = 2 into the first one.

Equation 1 2(1) + 2(2) = 2 + 4 = 6

We get 6, the same as the right side.

Now equation 2.

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