Solve 3x + 4y = 2, 2x + 3y = 1 using the inverse matrix Step by Step

Step by step

The question

Question: solve 3x + 4y = 2 and 2x + 3y = 1 using the inverse matrix.

3x + 4y = 2

We want the x and y that make both true.

2x + 3y = 1

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.

[3, 4; 2, 3] × [x; y] = [2; 1]

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 = |3, 4; 2, 3|

So D = 3 × 3 − 2 × 4.

D = 3 × 3 − 2 × 4

3 times 3 is 9.

2 times 4 is 8.

D = 9 − 8

9 minus 8 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, −4; −2, 3]

So A⁻¹ = rows 3, −4 and −2, 3.

A⁻¹ = [3, −4; −2, 3]

3 Multiply

X is A⁻¹ times B.

[x; y] = [3, −4; −2, 3] × [2; 1]

= [3 × 2 + (−4) × 1; −2 × 2 + 3 × 1]

Work them out: x = 2 and y = −1.

[x; y] = [2; −1]

So the answer is (2, −1).

(x, y) = (2, −1)

4 Check

Let's check in every equation.

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

Equation 1 3(2) + 4(−1) = 6 − 4 = 2

We get 2, the same as the right side.

Now equation 2.

Equation 2 2(2) + 3(−1) = 4 − 3 = 1

We get 1, 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