Solve 5x + 2y = 9, 2x + y = 4 using the inverse matrix Step by Step

Step by step

The question

Question: solve 5x + 2y = 9 and 2x + y = 4 using the inverse matrix.

5x + 2y = 9

We want the x and y that make both true.

2x + y = 4

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.

[5, 2; 2, 1] × [x; y] = [9; 4]

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 = |5, 2; 2, 1|

So D = 5 × 1 − 2 × 2.

D = 5 × 1 − 2 × 2

5 times 1 is 5.

2 times 2 is 4.

D = 5 − 4

5 minus 4 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 × [1, −2; −2, 5]

So A⁻¹ = rows 1, −2 and −2, 5.

A⁻¹ = [1, −2; −2, 5]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1, −2; −2, 5] × [9; 4]

= [1 × 9 + (−2) × 4; −2 × 9 + 5 × 4]

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 5(1) + 2(2) = 5 + 4 = 9

We get 9, the same as the right side.

Now equation 2.

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

We get 4, 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