Answer:
[tex]\fbox{\begin{minipage}{8em}(x, y) = (-2, 3)\end{minipage}}[/tex]
Step-by-step explanation:
Given:
4x + 5y = 7
3x - 2y = -12
Solve for:
x and y
Solution:
Step 1: Find the good approach
There are plenty of ways to solve system of linear equations. For a problem requiring solving only 2 variables, one of the good methods is trying to eliminate all components containing 1 variable, then solve for remaining variable only. Afterwards, substitute the variable found, back into one of original equations to work out the eliminated variable.
Step 2: Perform the calculation
Transform the original system into new one which is easier for eliminating:
4x + 5y = 7
3x - 2y = -12
<=>
3*(4x + 5y) = 3*7 (multiply both sides of 1st equation by 3)
4*(3x - 2y) = 4*(-12) (multiply both sides of 2nd equation by 4)
As you might see, we intentionally try to create the components containing x, that share the same coefficient ( 3 x 4 = 4 x 3 = 12), this would help the next eliminating step.
<=>
12x + 15y = 21
12x - 8y = -48
Now, we see that if we subtract 2nd equation from 1st equation, we will completely eliminate components containing x.
<=>
(12x - 12x) + (15y - (-8y) = 21 - (-48)
Then, we do the simplification:
<=>
23y = 69
We finally divide both sides of equation by 23, to work out y.
<=> (23/23)y = 69/23
<=> y = 3
Now we substitute y back into 1st original equation, we have:
4x + 5y = 7 or 4x + 5*3 = 7
<=> 4x + 15 = 7
We move all numbers to the right side, except the components containing x (notice the change of sign of number from + to -)
<=> 4x = 7 - 15
<=> 4x = -8
We finally divide both sides of equation by 4, to work out x.
<=> (4/4)x = -8/4
<=> x = -2
Now, we conclude the solution of this system of linear equation is:
(x, y) = (-2, 3)
Hope this helps!
:)
