Solve 5x − 5y = 15, 4x − 3y = 8 by elimination
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −1, y = −4 |
Step-by-step solution
9 steps-
1 The system, with the equations numbered\begin{aligned}5 x - 5 y &= 15 & (1)\\ 4 x - 3 y &= 8 & (2)\end{aligned}
-
2 Multiply equation (1) by 3 and equation (2) by 5 so the y-terms match
Elimination needs one unknown to have the same or opposite coefficient in both equations. Multiplying a whole equation by a number keeps it true, so the equations are scaled until the y-terms match.
\begin{aligned}15 x - 15 y &= 45 & (1)\\ 20 x - 15 y &= 40 & (2)\end{aligned} -
3 Subtract equation (2) from equation (1) to eliminate y
Adding or subtracting two true equations gives another true equation. Because the y-terms match, they cancel, leaving one equation in one unknown.
- 5 x = 5 -
4 Divide both sides by -5
Dividing both sides by -5 undoes the multiplication by -5, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{5}{-5} -
5 Simplifyx = -1
-
6 Substitute x = -1 into equation (2)
Once one unknown is known, putting its value into an earlier equation gives the other one.
- 3 y + 4 \left(-1\right) = 8 -
7 Add 4 to both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
- 3 y = 12Move the constant terms to the right.
-
8 Divide both sides by -3
Dividing both sides by -3 undoes the multiplication by -3, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
y = \frac{12}{-3} -
9 Simplifyy = -4
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 15 | 15 | ✓ |
| (2) | 8 | 8 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.