Linear Equations Worksheet for solving unknown variables.

Here's the step-by-step solution to all 10 linear equations from the worksheet.

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Problem 1:
Equation: $-\frac{9}{5}x = -45$

Solution:

Multiply both sides by 5 to eliminate denominator:
$$
-9x = -225
$$

Divide both sides by -9:
$$
x = \frac{-225}{-9} = 25
$$

✔ Answer: $x = 25$

---

Problem 2:
Equation: $\frac{x}{3} - \frac{x}{5} = 2$

Solution:

Find LCD of 3 and 5 → 15. Multiply entire equation by 15:
$$
15 \cdot \left( \frac{x}{3} - \frac{x}{5} \right) = 15 \cdot 2
$$
$$
5x - 3x = 30
$$
$$
2x = 30
$$
$$
x = 15
$$

✔ Answer: $x = 15$

---

Problem 3:
Equation: $\frac{4x + 5}{6} = \frac{7}{2}$

Solution:

Multiply both sides by 6:
$$
4x + 5 = 6 \cdot \frac{7}{2} = 21
$$

Subtract 5:
$$
4x = 16
$$

Divide by 4:
$$
x = 4
$$

✔ Answer: $x = 4$

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Problem 4:
Equation: $8 = 2(x - 5) + 6x$

Solution:

Distribute the 2:
$$
8 = 2x - 10 + 6x
$$

Combine like terms:
$$
8 = 8x - 10
$$

Add 10 to both sides:
$$
18 = 8x
$$

Divide by 8:
$$
x = \frac{18}{8} = \frac{9}{4}
$$

✔ Answer: $x = \frac{9}{4}$

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Problem 5:
Equation: $-(x + 2) = 2(3x - 4)$

Solution:

Distribute negatives and multiplication:
Left: $-x - 2$
Right: $6x - 8$

So:
$$
-x - 2 = 6x - 8
$$

Add x to both sides:
$$
-2 = 7x - 8
$$

Add 8:
$$
6 = 7x
$$

Divide by 7:
$$
x = \frac{6}{7}
$$

✔ Answer: $x = \frac{6}{7}$

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Problem 6:
Equation: $3 = 4(x - 2) + 5 - 3x$

Solution:

Distribute the 4:
$$
3 = 4x - 8 + 5 - 3x
$$

Combine like terms:
$$
3 = (4x - 3x) + (-8 + 5) = x - 3
$$

Add 3 to both sides:
$$
6 = x
$$

✔ Answer: $x = 6$

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Problem 7:
Equation: $\frac{2x - 1}{3} - \frac{3x}{4} = \frac{5}{6}$

Solution:

Find LCD of 3, 4, 6 → 12. Multiply entire equation by 12:
$$
12 \cdot \left( \frac{2x - 1}{3} - \frac{3x}{4} \right) = 12 \cdot \frac{5}{6}
$$

Simplify each term:
$$
4(2x - 1) - 3(3x) = 10
$$
$$
8x - 4 - 9x = 10
$$
$$
-x - 4 = 10
$$

Add 4:
$$
-x = 14
$$

Multiply by -1:
$$
x = -14
$$

✔ Answer: $x = -14$

---

Problem 8:
Equation: $-5x + 3 = 2x + 8$

Solution:

Move variables to one side, constants to other. Subtract 2x:
$$
-7x + 3 = 8
$$

Subtract 3:
$$
-7x = 5
$$

Divide by -7:
$$
x = -\frac{5}{7}
$$

✔ Answer: $x = -\frac{5}{7}$

---

Problem 9:
Equation: $4 = -(2x + 4)$

Solution:

Distribute the negative:
$$
4 = -2x - 4
$$

Add 4 to both sides:
$$
8 = -2x
$$

Divide by -2:
$$
x = -4
$$

✔ Answer: $x = -4$

---

Problem 10:
Equation: $\frac{7}{8}y - 6 = 8$

Solution:

Add 6 to both sides:
$$
\frac{7}{8}y = 14
$$

Multiply both sides by 8:
$$
7y = 112
$$

Divide by 7:
$$
y = 16
$$

✔ Answer: $y = 16$

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## ✔ Final Answers:

1. $x = 25$
2. $x = 15$
3. $x = 4$
4. $x = \frac{9}{4}$
5. $x = \frac{6}{7}$
6. $x = 6$
7. $x = -14$
8. $x = -\frac{5}{7}$
9. $x = -4$
10. $y = 16$

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