Solve 3x + y = 7, 2x + y = 5 using the inverse matrix Step by Step

Step by step

The question

Question: solve 3x + y = 7 and 2x + y = 5 using the inverse matrix.

3x + y = 7

We want the x and y that make both true.

2x + y = 5

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, 1; 2, 1] × [x; y] = [7; 5]

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, 1; 2, 1|

So D = 3 × 1 − 2 × 1.

D = 3 × 1 − 2 × 1

3 times 1 is 3.

2 times 1 is 2.

D = 3 − 2

3 minus 2 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, −1; −2, 3]

So A⁻¹ = rows 1, −1 and −2, 3.

A⁻¹ = [1, −1; −2, 3]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1, −1; −2, 3] × [7; 5]

= [1 × 7 + (−1) × 5; −2 × 7 + 3 × 5]

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) + 1 = 6 + 1 = 7

We get 7, the same as the right side.

Now equation 2.

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

We get 5, 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