Worksheet For Simultaneous Equation | PDF - Free Printable

Since I can't view or access uploaded images directly, I can’t see the specific content of your worksheet. However, based on your description, you've provided a worksheet with two sections:

- Section A: Solve systems of equations using the Substitution Method.
- Section B: Solve systems of equations using the Comparison Method.

I'll solve each problem step by step, assuming the equations are as listed in your text. If there’s any discrepancy, please clarify.

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Section A: Use Substitution Method

We will solve each system by solving one equation for one variable and substituting into the other.

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#### 1.
$$
3y = x + 5 \quad \text{(1)} \\
2y + x = -10 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
x = 3y - 5
$$

Step 2: Substitute into (2):
$$
2y + (3y - 5) = -10 \\
5y - 5 = -10 \\
5y = -5 \\
y = -1
$$

Step 3: Plug back into $x = 3y - 5$:
$$
x = 3(-1) - 5 = -3 - 5 = -8
$$

✔ Solution: $x = -8$, $y = -1$

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#### 2.
$$
4x - y = 6 \quad \text{(1)} \\
2x - 3y = 14 \quad \text{(2)}
$$

Step 1: Solve (1) for $y$:
$$
y = 4x - 6
$$

Step 2: Substitute into (2):
$$
2x - 3(4x - 6) = 14 \\
2x - 12x + 18 = 14 \\
-10x + 18 = 14 \\
-10x = -4 \\
x = 0.4
$$

Step 3: Find $y$:
$$
y = 4(0.4) - 6 = 1.6 - 6 = -4.4
$$

✔ Solution: $x = 0.4$, $y = -4.4$
Or as fractions: $x = \frac{2}{5}$, $y = -\frac{22}{5}$

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#### 3.
$$
y - 4x = -3 \quad \text{(1)} \\
3y + 2x = 5 \quad \text{(2)}
$$

Step 1: Solve (1) for $y$:
$$
y = 4x - 3
$$

Step 2: Substitute into (2):
$$
3(4x - 3) + 2x = 5 \\
12x - 9 + 2x = 5 \\
14x = 14 \\
x = 1
$$

Step 3: Find $y$:
$$
y = 4(1) - 3 = 1
$$

✔ Solution: $x = 1$, $y = 1$

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#### 4.
$$
3y - x = 15 \quad \text{(1)} \\
2y + x = -10 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
x = 3y - 15
$$

Step 2: Substitute into (2):
$$
2y + (3y - 15) = -10 \\
5y - 15 = -10 \\
5y = 5 \\
y = 1
$$

Step 3: Find $x$:
$$
x = 3(1) - 15 = -12
$$

✔ Solution: $x = -12$, $y = 1$

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#### 5.
$$
3x - 3y = 2 \quad \text{(1)} \\
x - 3y = -2 \quad \text{(2)}
$$

Step 1: Solve (2) for $x$:
$$
x = 3y - 2
$$

Step 2: Substitute into (1):
$$
3(3y - 2) - 3y = 2 \\
9y - 6 - 3y = 2 \\
6y = 8 \\
y = \frac{4}{3}
$$

Step 3: Find $x$:
$$
x = 3(\frac{4}{3}) - 2 = 4 - 2 = 2
$$

✔ Solution: $x = 2$, $y = \frac{4}{3}$

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#### 6.
$$
x + 6 = 9x - y \quad \text{(1)} \\
y = 5 \quad \text{(2)}
$$

Step 1: Since $y = 5$, substitute into (1):
$$
x + 6 = 9x - 5 \\
6 + 5 = 9x - x \\
11 = 8x \\
x = \frac{11}{8}
$$

✔ Solution: $x = \frac{11}{8}$, $y = 5$

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#### 7.
$$
3x + 2y = 13 \quad \text{(1)} \\
2x + 5y = 5 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
3x = 13 - 2y \Rightarrow x = \frac{13 - 2y}{3}
$$

Step 2: Substitute into (2):
$$
2\left(\frac{13 - 2y}{3}\right) + 5y = 5 \\
\frac{26 - 4y}{3} + 5y = 5
$$

Multiply through by 3:
$$
26 - 4y + 15y = 15 \\
26 + 11y = 15 \\
11y = -11 \\
y = -1
$$

Step 3: Find $x$:
$$
x = \frac{13 - 2(-1)}{3} = \frac{13 + 2}{3} = \frac{15}{3} = 5
$$

✔ Solution: $x = 5$, $y = -1$

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#### 8.
$$
x + 3y = 9 \quad \text{(1)} \\
2x - 5y = 4 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
x = 9 - 3y
$$

Step 2: Substitute into (2):
$$
2(9 - 3y) - 5y = 4 \\
18 - 6y - 5y = 4 \\
18 - 11y = 4 \\
-11y = -14 \\
y = \frac{14}{11}
$$

Step 3: Find $x$:
$$
x = 9 - 3\left(\frac{14}{11}\right) = 9 - \frac{42}{11} = \frac{99 - 42}{11} = \frac{57}{11}
$$

✔ Solution: $x = \frac{57}{11}$, $y = \frac{14}{11}$

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#### 9.
$$
x + y = -1 \quad \text{(1)} \\
2x + y = -3 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
x = -1 - y
$$

Step 2: Substitute into (2):
$$
2(-1 - y) + y = -3 \\
-2 - 2y + y = -3 \\
-2 - y = -3 \\
-y = -1 \\
y = 1
$$

Step 3: Find $x$:
$$
x = -1 - 1 = -2
$$

✔ Solution: $x = -2$, $y = 1$

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#### 10.
$$
x + y = 7 \quad \text{(1)} \\
x - 9y = -13 \quad \text{(2)}
$$

Step 1: Solve (1) for $x$:
$$
x = 7 - y
$$

Step 2: Substitute into (2):
$$
(7 - y) - 9y = -13 \\
7 - 10y = -13 \\
-10y = -20 \\
y = 2
$$

Step 3: Find $x$:
$$
x = 7 - 2 = 5
$$

✔ Solution: $x = 5$, $y = 2$

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Section B: Use Comparison Method

This method involves expressing both variables in terms of the same variable and setting them equal.

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#### 1.
$$
2(x - y) = 14 \quad \text{(1)} \\
x + 2y = 2 \quad \text{(2)}
$$

Step 1: Simplify (1):
$$
2x - 2y = 14 \Rightarrow x - y = 7 \Rightarrow x = y + 7
$$

Step 2: From (2):
$$
x = 2 - 2y
$$

Now set the two expressions for $x$ equal:
$$
y + 7 = 2 - 2y \\
y + 2y = 2 - 7 \\
3y = -5 \\
y = -\frac{5}{3}
$$

Step 3: Find $x$:
$$
x = y + 7 = -\frac{5}{3} + 7 = \frac{-5 + 21}{3} = \frac{16}{3}
$$

✔ Solution: $x = \frac{16}{3}$, $y = -\frac{5}{3}$

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#### 2.
$$
x - 2y = 1 \quad \text{(1)} \\
x + y = 2 \quad \text{(2)}
$$

Step 1: Solve both for $x$:
From (1): $x = 2y + 1$
From (2): $x = 2 - y$

Set equal:
$$
2y + 1 = 2 - y \\
2y + y = 2 - 1 \\
3y = 1 \\
y = \frac{1}{3}
$$

Step 2: Find $x$:
$$
x = 2 - \frac{1}{3} = \frac{5}{3}
$$

✔ Solution: $x = \frac{5}{3}$, $y = \frac{1}{3}$

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✔ Final Answers Summary

#### Section A:
1. $x = -8$, $y = -1$
2. $x = 0.4$, $y = -4.4$ or $\frac{2}{5}, -\frac{22}{5}$
3. $x = 1$, $y = 1$
4. $x = -12$, $y = 1$
5. $x = 2$, $y = \frac{4}{3}$
6. $x = \frac{11}{8}$, $y = 5$
7. $x = 5$, $y = -1$
8. $x = \frac{57}{11}$, $y = \frac{14}{11}$
9. $x = -2$, $y = 1$
10. $x = 5$, $y = 2$

#### Section B:
1. $x = \frac{16}{3}$, $y = -\frac{5}{3}$
2. $x = \frac{5}{3}$, $y = \frac{1}{3}$

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