Set of practice problems designed to test your skills in solving linear equations with one variable.

Problem Set: Linear Equations in One Variable

Below, I will solve each problem step by step.

---

#### Q1. Solve the equation:
\[
\frac{1}{x} + \frac{2}{x} = 3
\]

Solution:
1. Combine the fractions on the left-hand side:
\[
\frac{1}{x} + \frac{2}{x} = \frac{1 + 2}{x} = \frac{3}{x}
\]
2. The equation becomes:
\[
\frac{3}{x} = 3
\]
3. Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[
3 = 3x
\]
4. Divide both sides by 3:
\[
x = 1
\]

Answer:
\[
\boxed{x = 1}
\]

---

#### Q2. The two-thirds of a number increased by 9 equals 19. Find the number.

Solution:
1. Let the number be \( x \).
2. According to the problem, two-thirds of the number increased by 9 is equal to 19:
\[
\frac{2}{3}x + 9 = 19
\]
3. Subtract 9 from both sides:
\[
\frac{2}{3}x = 10
\]
4. Multiply both sides by 3 to eliminate the fraction:
\[
2x = 30
\]
5. Divide both sides by 2:
\[
x = 15
\]

Answer:
\[
\boxed{x = 15}
\]

---

#### Q3. Solve the equation:
\[
\frac{6}{2x - (3 - 4x)} = \frac{2}{3}
\]

Solution:
1. Simplify the denominator on the left-hand side:
\[
2x - (3 - 4x) = 2x - 3 + 4x = 6x - 3
\]
So the equation becomes:
\[
\frac{6}{6x - 3} = \frac{2}{3}
\]
2. Cross-multiply to eliminate the fractions:
\[
6 \cdot 3 = 2 \cdot (6x - 3)
\]
\[
18 = 12x - 6
\]
3. Add 6 to both sides:
\[
24 = 12x
\]
4. Divide both sides by 12:
\[
x = 2
\]

Answer:
\[
\boxed{x = 2}
\]

---

#### Q4. Solve the equation:
\[
\frac{2x}{3x + 1} = -3
\]

Solution:
1. Cross-multiply to eliminate the fraction:
\[
2x = -3(3x + 1)
\]
2. Distribute the \(-3\) on the right-hand side:
\[
2x = -9x - 3
\]
3. Add \(9x\) to both sides:
\[
2x + 9x = -3
\]
\[
11x = -3
\]
4. Divide both sides by 11:
\[
x = -\frac{3}{11}
\]

Answer:
\[
\boxed{x = -\frac{3}{11}}
\]

---

#### Q5. Solve the equation:
\[
5x - 3 = 3x - 5
\]

Solution:
1. Subtract \(3x\) from both sides:
\[
5x - 3x - 3 = -5
\]
\[
2x - 3 = -5
\]
2. Add 3 to both sides:
\[
2x = -2
\]
3. Divide both sides by 2:
\[
x = -1
\]

Answer:
\[
\boxed{x = -1}
\]

---

#### Q6. Solve the equation:
\[
\frac{5x - 7}{3x} = 2
\]

Solution:
1. Cross-multiply to eliminate the fraction:
\[
5x - 7 = 2 \cdot 3x
\]
\[
5x - 7 = 6x
\]
2. Subtract \(5x\) from both sides:
\[
-7 = 6x - 5x
\]
\[
-7 = x
\]

Answer:
\[
\boxed{x = -7}
\]

---

#### Q7. Solve the equation:
\[
\frac{x}{5} + 1 = \frac{1}{15}
\]

Solution:
1. Subtract 1 from both sides:
\[
\frac{x}{5} = \frac{1}{15} - 1
\]
2. Write 1 as \(\frac{15}{15}\):
\[
\frac{x}{5} = \frac{1}{15} - \frac{15}{15}
\]
\[
\frac{x}{5} = \frac{1 - 15}{15}
\]
\[
\frac{x}{5} = \frac{-14}{15}
\]
3. Multiply both sides by 5:
\[
x = 5 \cdot \frac{-14}{15}
\]
\[
x = \frac{-70}{15}
\]
4. Simplify the fraction:
\[
x = \frac{-14}{3}
\]

Answer:
\[
\boxed{x = -\frac{14}{3}}
\]

---

#### Q8. Solve the equation:
\[
x - 2x + 2 - \frac{16}{2}x = 3 - \frac{7}{2}x
\]

Solution:
1. Simplify the terms:
- On the left-hand side, \(\frac{16}{2}x = 8x\):
\[
x - 2x + 2 - 8x
\]
Combine like terms:
\[
(x - 2x - 8x) + 2 = -9x + 2
\]
- On the right-hand side, keep it as is:
\[
3 - \frac{7}{2}x
\]
So the equation becomes:
\[
-9x + 2 = 3 - \frac{7}{2}x
\]
2. Eliminate the fraction by multiplying every term by 2:
\[
2(-9x) + 2(2) = 2(3) - 2\left(\frac{7}{2}x\right)
\]
\[
-18x + 4 = 6 - 7x
\]
3. Add \(18x\) to both sides:
\[
4 = 6 - 7x + 18x
\]
\[
4 = 6 + 11x
\]
4. Subtract 6 from both sides:
\[
4 - 6 = 11x
\]
\[
-2 = 11x
\]
5. Divide both sides by 11:
\[
x = -\frac{2}{11}
\]

Answer:
\[
\boxed{x = -\frac{2}{11}}
\]

---

#### Q9. Solve the equation:
\[
\frac{x^2 + 4}{3x^2 + 7} = \frac{1}{2}
\]

Solution:
1. Cross-multiply to eliminate the fraction:
\[
2(x^2 + 4) = 1(3x^2 + 7)
\]
\[
2x^2 + 8 = 3x^2 + 7
\]
2. Subtract \(2x^2\) from both sides:
\[
8 = 3x^2 - 2x^2 + 7
\]
\[
8 = x^2 + 7
\]
3. Subtract 7 from both sides:
\[
8 - 7 = x^2
\]
\[
1 = x^2
\]
4. Take the square root of both sides:
\[
x = \pm 1
\]

Answer:
\[
\boxed{x = \pm 1}
\]

---

Final Answers:

1. \(\boxed{x = 1}\)
2. \(\boxed{x = 15}\)
3. \(\boxed{x = 2}\)
4. \(\boxed{x = -\frac{3}{11}}\)
5. \(\boxed{x = -1}\)
6. \(\boxed{x = -7}\)
7. \(\boxed{x = -\frac{14}{3}}\)
8. \(\boxed{x = -\frac{2}{11}}\)
9. \(\boxed{x = \pm 1}\)