Solve 5x + 3y = 1, 3x + 2y = 0 using the inverse matrix Step by Step

Step by step

The question

Question: solve 5x + 3y = 1 and 3x + 2y = 0 using the inverse matrix.

5x + 3y = 1

We want the x and y that make both true.

3x + 2y = 0

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, 3; 3, 2] × [x; y] = [1; 0]

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, 3; 3, 2|

So D = 5 × 2 − 3 × 3.

D = 5 × 2 − 3 × 3

5 times 2 is 10.

3 times 3 is 9.

D = 10 − 9

10 minus 9 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; −3, 5]

So A⁻¹ = rows 2, −3 and −3, 5.

A⁻¹ = [2, −3; −3, 5]

3 Multiply

X is A⁻¹ times B.

[x; y] = [2, −3; −3, 5] × [1; 0]

= [2 × 1 + (−3) × 0; −3 × 1 + 5 × 0]

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 5(2) + 3(−3) = 10 − 9 = 1

We get 1, the same as the right side.

Now equation 2.

Equation 2 3(2) + 2(−3) = 6 − 6 = 0

We get 0, 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