Solve 3x − 4y = 16, x − 5y = 9 by elimination
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 4, y = −1 |
Step-by-step solution
7 steps-
1 The system, with the equations numbered\begin{aligned}3 x - 4 y &= 16 & (1)\\ x - 5 y &= 9 & (2)\end{aligned}
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2 Multiply equation (2) by 3 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}3 x - 4 y &= 16 & (1)\\ 3 x - 15 y &= 27 & (2)\end{aligned} -
3 Subtract equation (2) from equation (1) 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.
11 y = -11 -
4 Divide both sides by 11
Dividing both sides by 11 undoes the multiplication by 11, 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{-11}{11} -
5 Simplifyy = -1
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6 Substitute y = -1 into equation (2)
Once one unknown is known, putting its value into an earlier equation gives the other one.
x - 5 \left(-1\right) = 9 -
7 Subtract 5 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.
x = 4Move the constant terms to the right.
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 16 | 16 | ✓ |
| (2) | 9 | 9 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.