Solve 4x + 3y = 2, 2x - y = 6 by Cramer's rule Step by Step

Step by step

The question

Question: solve 4x + 3y = 2 and 2x − y = 6 by Cramer's rule.

4x + 3y = 2①

We call them ① and ②.

We want the x and y that make both true.

2x − y = 6②

Cramer's rule uses three determinants: D, D x and D y.

Then x is D x over D, and y is D y over D.

x = D_xD, y = D_yD

1 Find D

D: the numbers in front of x and y, in the same places as in the equations.

D = |4, 3; 2, −1|

So D = 4 × (−1) − 2 × 3.

D = 4 × (−1) − 2 × 3

4 times (−1) is −4.

2 times 3 is 6.

D = −4 − 6

−4 minus 6 is −10.

So D = −10.

D = −10

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 2 and 6.

D_x = |2, 3; 6, −1|

Careful!

For D x, we replace the x column: the first one.

The y column stays.

Careful: not the y column replaced ✗

So D x = 2 × (−1) − 6 × 3.

D_x = 2 × (−1) − 6 × 3

2 times (−1) is −2.

6 times 3 is 18.

D_x = −2 − 18

−2 minus 18 is −20.

So D x = −20.

D_x = −20

3 Find Dy

D y: now the y column is replaced by 2 and 6.

D_y = |4, 2; 2, 6|

So D y = 4 × 6 − 2 × 2.

D_y = 4 × 6 − 2 × 2

4 times 6 is 24.

2 times 2 is 4.

D_y = 24 − 4

24 minus 4 is 20.

So D y = 20.

D_y = 20

4 Divide

x is D x over D: −20 over (−10).

x = D_xD = −20−10

−20 divided by (−10) is 2.

So x = 2.

x = 2

y is D y over D: 20 over (−10).

y = D_yD = 20−10

20 divided by (−10) is −2.

So y = −2.

y = −2

So x = 2 and y = −2, both at the same time.

We write it as (2, −2).

(x, y) = (2, −2)

5 Check

Let's check in every equation.

We put x = 2, y = −2 into ①.

① 4(2) + 3(−2) = 8 − 6 = 2

We get 2, the same as the right side.

Now ②.

② 2(2) − (−2) = 4 + 2 = 6

We get 6, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: find D.

Then: find Dx.

Then: find Dy.

Then: divide.

Last: check.

Answer