Worksheet # 1: Solve the Linear Equations Using the Substitution ... - Free Printable

Here are the step-by-step solutions for each problem on the worksheet using the substitution method.

1) $y = -3x$ and $y = x - 8$
* Since both equations equal $y$, set them equal to each other:
$-3x = x - 8$
* Subtract $x$ from both sides:
$-4x = -8$
* Divide by $-4$:
$x = 2$
* Plug $x = 2$ into the first equation ($y = -3x$):
$y = -3(2)$
$y = -6$
* Solution: $(2, -6)$

2) $y = 3x$ and $y = -8x$
* Set them equal:
$3x = -8x$
* Add $8x$ to both sides:
$11x = 0$
* Divide by $11$:
$x = 0$
* Plug $x = 0$ into the first equation:
$y = 3(0)$
$y = 0$
* Solution: $(0, 0)$

3) $y = -2x$ and $y = 4x + 10$
* Set them equal:
$-2x = 4x + 10$
* Subtract $4x$ from both sides:
$-6x = 10$
* Divide by $-6$:
$x = -\frac{10}{6}$ which simplifies to $-\frac{5}{3}$
* Plug $x = -\frac{5}{3}$ into the first equation:
$y = -2(-\frac{5}{3})$
$y = \frac{10}{3}$
* Solution: $(-\frac{5}{3}, \frac{10}{3})$

4) $y = -7x$ and $y = -4x - 12$
* Set them equal:
$-7x = -4x - 12$
* Add $4x$ to both sides:
$-3x = -12$
* Divide by $-3$:
$x = 4$
* Plug $x = 4$ into the first equation:
$y = -7(4)$
$y = -28$
* Solution: $(4, -28)$

5) $y = 3x$ and $y = 2x - 7$
* Set them equal:
$3x = 2x - 7$
* Subtract $2x$ from both sides:
$x = -7$
* Plug $x = -7$ into the first equation:
$y = 3(-7)$
$y = -21$
* Solution: $(-7, -21)$

6) $y = 2x + 3$ and $y = 3$
* Since we know $y = 3$, plug that value directly into the first equation:
$3 = 2x + 3$
* Subtract $3$ from both sides:
$0 = 2x$
* Divide by $2$:
$x = 0$
* We already know $y = 3$.
* Solution: $(0, 3)$

7) $y = 6x + 22$ and $y = -8$
* Since we know $y = -8$, plug that into the first equation:
$-8 = 6x + 22$
* Subtract $22$ from both sides:
$-30 = 6x$
* Divide by $6$:
$x = -5$
* We already know $y = -8$.
* Solution: $(-5, -8)$

8) $y = 2x - 5$ and $y = x$
* Substitute $x$ for $y$ in the first equation:
$x = 2x - 5$
* Subtract $2x$ from both sides:
$-x = -5$
* Multiply by $-1$:
$x = 5$
* Since $y = x$, then $y = 5$.
* Solution: $(5, 5)$

9) $y = 4x + 10$ and $y = -6$
* Plug $y = -6$ into the first equation:
$-6 = 4x + 10$
* Subtract $10$ from both sides:
$-16 = 4x$
* Divide by $4$:
$x = -4$
* We already know $y = -6$.
* Solution: $(-4, -6)$

10) $y = 8$ and $y = -2x + 22$
* Plug $y = 8$ into the second equation:
$8 = -2x + 22$
* Subtract $22$ from both sides:
$-14 = -2x$
* Divide by $-2$:
$x = 7$
* We already know $y = 8$.
* Solution: $(7, 8)$

11) $y = 4$ and $y = 4x - 24$
* Plug $y = 4$ into the second equation:
$4 = 4x - 24$
* Add $24$ to both sides:
$28 = 4x$
* Divide by $4$:
$x = 7$
* We already know $y = 4$.
* Solution: $(7, 4)$

12) $y = -6x$ and $y = -3x$
* Set them equal:
$-6x = -3x$
* Add $6x$ to both sides:
$0 = 3x$
* Divide by $3$:
$x = 0$
* Plug $x = 0$ into the first equation:
$y = -6(0)$
$y = 0$
* Solution: $(0, 0)$

Final Answer:
1) (2, -6)
2) (0, 0)
3) (-5/3, 10/3)
4) (4, -28)
5) (-7, -21)
6) (0, 3)
7) (-5, -8)
8) (5, 5)
9) (-4, -6)
10) (7, 8)
11) (7, 4)
12) (0, 0)