Math worksheet featuring simultaneous equations to solve using substitution and comparison methods.

Here are the step-by-step solutions for the problems on your worksheet.

Part A: Substitution Method

*Goal: Solve one equation for $x$ or $y$, then plug that into the other equation.*

1. $3y - x = 5$; $2y + x = -10$
* From the first equation, solve for $x$: $x = 3y - 5$.
* Substitute this into the second equation: $2y + (3y - 5) = -10$.
* Combine terms: $5y - 5 = -10$.
* Add 5 to both sides: $5y = -5$.
* Divide by 5: $y = -1$.
* Plug $y$ back in to find $x$: $x = 3(-1) - 5 = -3 - 5 = -8$.
* Answer: $x = -8, y = -1$

2. $4x - y = 8$; $2x - 3y = 14$
* From the first equation, solve for $y$: $y = 4x - 8$.
* Substitute into the second equation: $2x - 3(4x - 8) = 14$.
* Expand: $2x - 12x + 24 = 14$.
* Simplify: $-10x + 24 = 14$.
* Subtract 24: $-10x = -10$.
* Divide by -10: $x = 1$.
* Find $y$: $y = 4(1) - 8 = -4$.
* Answer: $x = 1, y = -4$

3. $y - 4x = -3$; $3y + 2x = 5$
* From the first equation, solve for $y$: $y = 4x - 3$.
* Substitute into the second equation: $3(4x - 3) + 2x = 5$.
* Expand: $12x - 9 + 2x = 5$.
* Combine terms: $14x - 9 = 5$.
* Add 9: $14x = 14$.
* Divide by 14: $x = 1$.
* Find $y$: $y = 4(1) - 3 = 1$.
* Answer: $x = 1, y = 1$

4. $3y - x = 15$; $2y + x = -10$
* From the first equation, solve for $x$: $x = 3y - 15$.
* Substitute into the second equation: $2y + (3y - 15) = -10$.
* Combine terms: $5y - 15 = -10$.
* Add 15: $5y = 5$.
* Divide by 5: $y = 1$.
* Find $x$: $x = 3(1) - 15 = -12$.
* Answer: $x = -12, y = 1$

5. $3x - 3y = 2$; $x - 3y = -2$
* From the second equation, solve for $x$: $x = 3y - 2$.
* Substitute into the first equation: $3(3y - 2) - 3y = 2$.
* Expand: $9y - 6 - 3y = 2$.
* Combine terms: $6y - 6 = 2$.
* Add 6: $6y = 8$.
* Divide by 6: $y = \frac{8}{6} = \frac{4}{3}$.
* Find $x$: $x = 3(\frac{4}{3}) - 2 = 4 - 2 = 2$.
* Answer: $x = 2, y = \frac{4}{3}$

6. $x + y = 9$; $x - y = 5$
* From the first equation, solve for $x$: $x = 9 - y$.
* Substitute into the second equation: $(9 - y) - y = 5$.
* Simplify: $9 - 2y = 5$.
* Subtract 9: $-2y = -4$.
* Divide by -2: $y = 2$.
* Find $x$: $x = 9 - 2 = 7$.
* Answer: $x = 7, y = 2$

7. $3x + 2y = 13$; $2x + 5y = 5$
* From the first equation, solve for $2y$: $2y = 13 - 3x$, so $y = \frac{13 - 3x}{2}$.
* Substitute into the second equation: $2x + 5(\frac{13 - 3x}{2}) = 5$.
* Multiply everything by 2 to clear the fraction: $4x + 5(13 - 3x) = 10$.
* Expand: $4x + 65 - 15x = 10$.
* Combine terms: $-11x + 65 = 10$.
* Subtract 65: $-11x = -55$.
* Divide by -11: $x = 5$.
* Find $y$: $3(5) + 2y = 13 \rightarrow 15 + 2y = 13 \rightarrow 2y = -2 \rightarrow y = -1$.
* Answer: $x = 5, y = -1$

8. $x + 3y = 9$; $2x - y = 4$
* From the first equation, solve for $x$: $x = 9 - 3y$.
* Substitute into the second equation: $2(9 - 3y) - y = 4$.
* Expand: $18 - 6y - y = 4$.
* Combine terms: $18 - 7y = 4$.
* Subtract 18: $-7y = -14$.
* Divide by -7: $y = 2$.
* Find $x$: $x = 9 - 3(2) = 9 - 6 = 3$.
* Answer: $x = 3, y = 2$

9. $x + y = -1$; $2x + y = -3$
* From the first equation, solve for $y$: $y = -1 - x$.
* Substitute into the second equation: $2x + (-1 - x) = -3$.
* Simplify: $x - 1 = -3$.
* Add 1: $x = -2$.
* Find $y$: $y = -1 - (-2) = -1 + 2 = 1$.
* Answer: $x = -2, y = 1$

10. $x + y = 7$; $x - 9y = -13$
* From the first equation, solve for $x$: $x = 7 - y$.
* Substitute into the second equation: $(7 - y) - 9y = -13$.
* Simplify: $7 - 10y = -13$.
* Subtract 7: $-10y = -20$.
* Divide by -10: $y = 2$.
* Find $x$: $x = 7 - 2 = 5$.
* Answer: $x = 5, y = 2$

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Part B: Comparison Method

*Goal: Rearrange both equations to make them equal to the same variable (like $x = ...$), then set them equal to each other.*

1. $2(x - y) = 14$; $x + 2y = -2$
* First, simplify the first equation by dividing by 2: $x - y = 7$.
* Make both equations equal to $x$:
* Eq 1: $x = y + 7$
* Eq 2: $x = -2 - 2y$
* Set them equal: $y + 7 = -2 - 2y$.
* Add $2y$ to both sides: $3y + 7 = -2$.
* Subtract 7: $3y = -9$.
* Divide by 3: $y = -3$.
* Find $x$: $x = -3 + 7 = 4$.
* Answer: $x = 4, y = -3$

2. $x - 2y = 1$; $x + y = 2$
* Make both equations equal to $x$:
* Eq 1: $x = 1 + 2y$
* Eq 2: $x = 2 - y$
* Set them equal: $1 + 2y = 2 - y$.
* Add $y$ to both sides: $1 + 3y = 2$.
* Subtract 1: $3y = 1$.
* Divide by 3: $y = \frac{1}{3}$.
* Find $x$: $x = 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3}$.
* Answer: $x = \frac{5}{3}, y = \frac{1}{3}$

Final Answer:
Part A:
1. $x = -8, y = -1$
2. $x = 1, y = -4$
3. $x = 1, y = 1$
4. $x = -12, y = 1$
5. $x = 2, y = \frac{4}{3}$
6. $x = 7, y = 2$
7. $x = 5, y = -1$
8. $x = 3, y = 2$
9. $x = -2, y = 1$
10. $x = 5, y = 2$

Part B:
1. $x = 4, y = -3$
2. $x = \frac{5}{3}, y = \frac{1}{3}$