Solve 5x + 3y = 8, −4x − 4y = −16
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −2, y = 6 |
Step-by-step solution
9 steps-
1 The system, with the equations numbered\begin{aligned}5 x + 3 y &= 8 & (1)\\ - 4 x - 4 y &= -16 & (2)\end{aligned}
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2 Multiply equation (1) by 4 and equation (2) by 3 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}20 x + 12 y &= 32 & (1)\\ - 12 x - 12 y &= -48 & (2)\end{aligned} -
3 Add the two equations 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.
8 x = -16 -
4 Divide both sides by 8
Dividing both sides by 8 undoes the multiplication by 8, 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{-16}{8} -
5 Simplifyx = -2
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6 Substitute x = -2 into equation (1)
Once one unknown is known, putting its value into an earlier equation gives the other one.
3 y + 5 \left(-2\right) = 8 -
7 Add 10 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 = 18Move the constant terms to the right.
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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{18}{3} -
9 Simplifyy = 6
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 8 | 8 | ✓ |
| (2) | −16 | −16 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.