Solve −5x + y = 11, −x − 3y = −17 by substitution
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −1, y = 6 |
Step-by-step solution
10 steps-
1 The system, with the equations numbered\begin{aligned}y - 5 x &= 11 & (1)\\ - x - 3 y &= -17 & (2)\end{aligned}
-
2 Solve equation (1) for y
Substitution starts by solving one equation for one unknown, ideally one with coefficient 1 or −1. Then y can be replaced in the other equation.
y - 5 x = 11 -
3 Add 5x 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.
y = 5 x + 11Move the constant terms to the right.
-
4 Substitute y = 5x + 11 into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- x - 3 \left(5 x + 11\right) = -17Now there is a single equation in x.
-
5 Distribute: multiply each term inside the brackets
The distributive property, a(b + c) = ab + ac: the number in front of a bracket multiplies every term inside it, not just the first. A minus in front works like −1, so it flips the sign of every term inside.
- x - 15 x - 33 = -17A minus sign in front of a bracket changes the sign of every term inside.
-
6 Combine like terms- 16 x - 33 = -17
-
7 Add 33 to both sides- 16 x = 16
Move the constant terms to the right.
-
8 Divide both sides by -16
Dividing both sides by -16 undoes the multiplication by -16, 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}{-16} -
9 Simplifyx = -1
-
10 Back-substitute x = -1 to find y
Once one unknown is known, putting its value into an earlier equation gives the other one.
y = 11 + 5 \left(-1\right) = 6
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 11 | 11 | ✓ |
| (2) | −17 | −17 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.