Solve 4x + 2y = 10, x + y = 3 using the inverse matrix Step by Step

Step by step

The question

Question: solve 4x + 2y = 10 and x + y = 3 using the inverse matrix.

4x + 2y = 10

We want the x and y that make both true.

x + y = 3

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.

[4, 2; 1, 1] × [x; y] = [10; 3]

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 = |4, 2; 1, 1|

So D = 4 × 1 − 1 × 2.

D = 4 × 1 − 1 × 2

4 times 1 is 4.

1 times 2 is 2.

D = 4 − 2

4 minus 2 is 2.

So D = 2.

D = 2

Swap the top left and bottom right.

Change the signs of the other two.

Then times 1 over D.

A⁻¹ = 12 × [1, −2; −1, 4]

So A⁻¹ = rows 1/2, −1 and −1/2, 2.

A⁻¹ = [1 over 2, −1; −1 over 2, 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1 over 2, −1; −1 over 2, 2] × [10; 3]

= [1 over 2 × 10 + (−1) × 3; −1 over 2 × 10 + 2 × 3]

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 4(2) + 2(1) = 8 + 2 = 10

We get 10, the same as the right side.

Now equation 2.

Equation 2 2 + 1 = 3

We get 3, 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