Solve −5x − 4y = 17, 2x + 5y = 0 by elimination
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −5, y = 2 |
Step-by-step solution
9 steps-
1 The system, with the equations numbered\begin{aligned}- 5 x - 4 y &= 17 & (1)\\ 2 x + 5 y &= 0 & (2)\end{aligned}
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2 Multiply equation (1) by 2 and equation (2) by 5 so the x-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 x-terms match.
\begin{aligned}- 10 x - 8 y &= 34 & (1)\\ 10 x + 25 y &= 0 & (2)\end{aligned} -
3 Add the two equations to eliminate x
Adding or subtracting two true equations gives another true equation. Because the x-terms match, they cancel, leaving one equation in one unknown.
17 y = 34 -
4 Divide both sides by 17
Dividing both sides by 17 undoes the multiplication by 17, 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{34}{17} -
5 Simplifyy = 2
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6 Substitute y = 2 into equation (2)
Once one unknown is known, putting its value into an earlier equation gives the other one.
2 x + 5 \cdot 2 = 0 -
7 Subtract 10 from 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.
2 x = -10Move the constant terms to the right.
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8 Divide both sides by 2
Dividing both sides by 2 undoes the multiplication by 2, 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{-10}{2} -
9 Simplifyx = -5
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 17 | 17 | ✓ |
| (2) | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.