imath - more exercises, re. solution of linear equations ... - Free Printable

Problem Set Solution

We will solve each equation step by step. Let's go through them one by one.

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#### 1. $\frac{5a}{3} - 2 = \frac{a}{4} + 15$

1. Eliminate the fractions by finding a common denominator (12):
$$
12 \left( \frac{5a}{3} - 2 \right) = 12 \left( \frac{a}{4} + 15 \right)
$$
$$
12 \cdot \frac{5a}{3} - 12 \cdot 2 = 12 \cdot \frac{a}{4} + 12 \cdot 15
$$
$$
4 \cdot 5a - 24 = 3a + 180
$$
$$
20a - 24 = 3a + 180
$$

2. Combine like terms:
$$
20a - 3a = 180 + 24
$$
$$
17a = 204
$$

3. Solve for \(a\):
$$
a = \frac{204}{17} = 12
$$

Answer:
$$
\boxed{12}
$$

---

#### 2. $\frac{x}{3} + \frac{x}{6} - 1 = \frac{5x}{12}$

1. Eliminate the fractions by finding a common denominator (12):
$$
12 \left( \frac{x}{3} + \frac{x}{6} - 1 \right) = 12 \left( \frac{5x}{12} \right)
$$
$$
12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{6} - 12 \cdot 1 = 12 \cdot \frac{5x}{12}
$$
$$
4x + 2x - 12 = 5x
$$
$$
6x - 12 = 5x
$$

2. Combine like terms:
$$
6x - 5x = 12
$$
$$
x = 12
$$

Answer:
$$
\boxed{12}
$$

---

#### 3. $\frac{2a}{5} - \frac{7}{5} = \frac{3a}{4} - \frac{5}{4}$

1. Eliminate the fractions by finding a common denominator (20):
$$
20 \left( \frac{2a}{5} - \frac{7}{5} \right) = 20 \left( \frac{3a}{4} - \frac{5}{4} \right)
$$
$$
20 \cdot \frac{2a}{5} - 20 \cdot \frac{7}{5} = 20 \cdot \frac{3a}{4} - 20 \cdot \frac{5}{4}
$$
$$
4 \cdot 2a - 4 \cdot 7 = 5 \cdot 3a - 5 \cdot 5
$$
$$
8a - 28 = 15a - 25
$$

2. Combine like terms:
$$
8a - 15a = -25 + 28
$$
$$
-7a = 3
$$

3. Solve for \(a\):
$$
a = -\frac{3}{7}
$$

Answer:
$$
\boxed{-\frac{3}{7}}
$$

---

#### 4. $\frac{x}{2} - \frac{x}{3} = \frac{x}{4} - 1$

1. Eliminate the fractions by finding a common denominator (12):
$$
12 \left( \frac{x}{2} - \frac{x}{3} \right) = 12 \left( \frac{x}{4} - 1 \right)
$$
$$
12 \cdot \frac{x}{2} - 12 \cdot \frac{x}{3} = 12 \cdot \frac{x}{4} - 12 \cdot 1
$$
$$
6x - 4x = 3x - 12
$$
$$
2x = 3x - 12
$$

2. Combine like terms:
$$
2x - 3x = -12
$$
$$
-x = -12
$$

3. Solve for \(x\):
$$
x = 12
$$

Answer:
$$
\boxed{12}
$$

---

#### 5. $\frac{3x}{4} + \frac{1}{2} = \frac{x}{5} + \frac{8}{5}$

1. Eliminate the fractions by finding a common denominator (20):
$$
20 \left( \frac{3x}{4} + \frac{1}{2} \right) = 20 \left( \frac{x}{5} + \frac{8}{5} \right)
$$
$$
20 \cdot \frac{3x}{4} + 20 \cdot \frac{1}{2} = 20 \cdot \frac{x}{5} + 20 \cdot \frac{8}{5}
$$
$$
5 \cdot 3x + 10 \cdot 1 = 4 \cdot x + 4 \cdot 8
$$
$$
15x + 10 = 4x + 32
$$

2. Combine like terms:
$$
15x - 4x = 32 - 10
$$
$$
11x = 22
$$

3. Solve for \(x\):
$$
x = \frac{22}{11} = 2
$$

Answer:
$$
\boxed{2}
$$

---

#### 6. $\frac{x}{4} - \frac{5}{12} = \frac{x}{2} - \frac{x}{3}$

1. Eliminate the fractions by finding a common denominator (12):
$$
12 \left( \frac{x}{4} - \frac{5}{12} \right) = 12 \left( \frac{x}{2} - \frac{x}{3} \right)
$$
$$
12 \cdot \frac{x}{4} - 12 \cdot \frac{5}{12} = 12 \cdot \frac{x}{2} - 12 \cdot \frac{x}{3}
$$
$$
3x - 5 = 6x - 4x
$$
$$
3x - 5 = 2x
$$

2. Combine like terms:
$$
3x - 2x = 5
$$
$$
x = 5
$$

Answer:
$$
\boxed{5}
$$

---

#### 7. $\frac{x}{5} = \frac{3}{35} + \frac{x+1}{7}$

1. Eliminate the fractions by finding a common denominator (35):
$$
35 \left( \frac{x}{5} \right) = 35 \left( \frac{3}{35} + \frac{x+1}{7} \right)
$$
$$
35 \cdot \frac{x}{5} = 35 \cdot \frac{3}{35} + 35 \cdot \frac{x+1}{7}
$$
$$
7x = 3 + 5(x + 1)
$$
$$
7x = 3 + 5x + 5
$$
$$
7x = 5x + 8
$$

2. Combine like terms:
$$
7x - 5x = 8
$$
$$
2x = 8
$$

3. Solve for \(x\):
$$
x = \frac{8}{2} = 4
$$

Answer:
$$
\boxed{4}
$$

---

#### 8. $\frac{2y-1}{3} + 3 = y$

1. Eliminate the fraction by multiplying through by 3:
$$
3 \left( \frac{2y-1}{3} + 3 \right) = 3 \cdot y
$$
$$
(2y - 1) + 9 = 3y
$$
$$
2y - 1 + 9 = 3y
$$
$$
2y + 8 = 3y
$$

2. Combine like terms:
$$
2y - 3y = -8
$$
$$
-y = -8
$$

3. Solve for \(y\):
$$
y = 8
$$

Answer:
$$
\boxed{8}
$$

---

#### 9. $\frac{5x+2}{3} + \frac{x}{5} = \frac{3x-5}{15} + x$

1. Eliminate the fractions by finding a common denominator (15):
$$
15 \left( \frac{5x+2}{3} + \frac{x}{5} \right) = 15 \left( \frac{3x-5}{15} + x \right)
$$
$$
15 \cdot \frac{5x+2}{3} + 15 \cdot \frac{x}{5} = 15 \cdot \frac{3x-5}{15} + 15 \cdot x
$$
$$
5(5x + 2) + 3x = (3x - 5) + 15x
$$
$$
25x + 10 + 3x = 3x - 5 + 15x
$$
$$
28x + 10 = 18x - 5
$$

2. Combine like terms:
$$
28x - 18x = -5 - 10
$$
$$
10x = -15
$$

3. Solve for \(x\):
$$
x = \frac{-15}{10} = -\frac{3}{2}
$$

Answer:
$$
\boxed{-\frac{3}{2}}
$$

---

#### 10. $\frac{2y-1}{3} + 3 = y$

This is the same as problem 8. The solution is:
$$
\boxed{8}
$$

---

#### 11. $\frac{1}{2} + \frac{x-1}{3} = \frac{x}{2}$

1. Eliminate the fractions by finding a common denominator (6):
$$
6 \left( \frac{1}{2} + \frac{x-1}{3} \right) = 6 \left( \frac{x}{2} \right)
$$
$$
6 \cdot \frac{1}{2} + 6 \cdot \frac{x-1}{3} = 6 \cdot \frac{x}{2}
$$
$$
3 + 2(x - 1) = 3x
$$
$$
3 + 2x - 2 = 3x
$$
$$
2x + 1 = 3x
$$

2. Combine like terms:
$$
2x - 3x = -1
$$
$$
-x = -1
$$

3. Solve for \(x\):
$$
x = 1
$$

Answer:
$$
\boxed{1}
$$

---

#### 12. $x - \frac{x-1}{2} = 0$

1. Eliminate the fraction by multiplying through by 2:
$$
2 \left( x - \frac{x-1}{2} \right) = 2 \cdot 0
$$
$$
2x - (x - 1) = 0
$$
$$
2x - x + 1 = 0
$$
$$
x + 1 = 0
$$

2. Solve for \(x\):
$$
x = -1
$$

Answer:
$$
\boxed{-1}
$$

---

#### 13. $\frac{4x}{3} - \frac{3x-4}{6} = 5 - \frac{x-2}{2}$

1. Eliminate the fractions by finding a common denominator (6):
$$
6 \left( \frac{4x}{3} - \frac{3x-4}{6} \right) = 6 \left( 5 - \frac{x-2}{2} \right)
$$
$$
6 \cdot \frac{4x}{3} - 6 \cdot \frac{3x-4}{6} = 6 \cdot 5 - 6 \cdot \frac{x-2}{2}
$$
$$
2 \cdot 4x - (3x - 4) = 30 - 3(x - 2)
$$
$$
8x - 3x + 4 = 30 - 3x + 6
$$
$$
5x + 4 = 36 - 3x
$$

2. Combine like terms:
$$
5x + 3x = 36 - 4
$$
$$
8x = 32
$$

3. Solve for \(x\):
$$
x = \frac{32}{8} = 4
$$

Answer:
$$
\boxed{4}
$$

---

#### 14. $\frac{x+1}{2} + \frac{x+2}{3} - \frac{x+3}{4} = 2$

1. Eliminate the fractions by finding a common denominator (12):
$$
12 \left( \frac{x+1}{2} + \frac{x+2}{3} - \frac{x+3}{4} \right) = 12 \cdot 2
$$
$$
12 \cdot \frac{x+1}{2} + 12 \cdot \frac{x+2}{3} - 12 \cdot \frac{x+3}{4} = 24
$$
$$
6(x + 1) + 4(x + 2) - 3(x + 3) = 24
$$
$$
6x + 6 + 4x + 8 - 3x - 9 = 24
$$
$$
7x + 5 = 24
$$

2. Combine like terms:
$$
7x = 24 - 5
$$
$$
7x = 19
$$

3. Solve for \(x\):
$$
x = \frac{19}{7}
$$

Answer:
$$
\boxed{\frac{19}{7}}
$$

---

#### 15. $\frac{5x-4}{3} - \frac{4x-3}{2} - \frac{3x-2}{1} = 0$

1. Eliminate the fractions by finding a common denominator (6):
$$
6 \left( \frac{5x-4}{3} - \frac{4x-3}{2} - \frac{3x-2}{1} \right) = 6 \cdot 0
$$
$$
6 \cdot \frac{5x-4}{3} - 6 \cdot \frac{4x-3}{2} - 6 \cdot \frac{3x-2}{1} = 0
$$
$$
2(5x - 4) - 3(4x - 3) - 6(3x - 2) = 0
$$
$$
10x - 8 - 12x + 9 - 18x + 12 = 0
$$
$$
10x - 12x - 18x - 8 + 9 + 12 = 0
$$
$$
-20x + 13 = 0
$$

2. Combine like terms:
$$
-20x = -13
$$

3. Solve for \(x\):
$$
x = \frac{13}{20}
$$

Answer:
$$
\boxed{\frac{13}{20}}
$$

---

#### 16. $\frac{3(2x+7)}{5} - 3 = \frac{5x-2}{3}$

1. Eliminate the fractions by finding a common denominator (15):
$$
15 \left( \frac{3(2x+7)}{5} - 3 \right) = 15 \left( \frac{5x-2}{3} \right)
$$
$$
15 \cdot \frac{3(2x+7)}{5} - 15 \cdot 3 = 15 \cdot \frac{5x-2}{3}
$$
$$
3 \cdot 3(2x + 7) - 45 = 5(5x - 2)
$$
$$
9(2x + 7) - 45 = 25x - 10
$$
$$
18x + 63 - 45 = 25x - 10
$$
$$
18x + 18 = 25x - 10
$$

2. Combine like terms:
$$
18x - 25x = -10 - 18
$$
$$
-7x = -28
$$

3. Solve for \(x\):
$$
x = \frac{-28}{-7} = 4
$$

Answer:
$$
\boxed{4}
$$

---

#### 17. $\frac{2x}{a} + \frac{b-a}{a} = \frac{2b}{a} \left( \frac{2x}{a} - \frac{b}{a} \right)$

1. Simplify the left-hand side:
$$
\frac{2x}{a} + \frac{b-a}{a} = \frac{2x + b - a}{a}
$$

2. Simplify the right-hand side:
$$
\frac{2b}{a} \left( \frac{2x}{a} - \frac{b}{a} \right) = \frac{2b}{a} \cdot \frac{2x - b}{a} = \frac{2b(2x - b)}{a^2} = \frac{4bx - 2b^2}{a^2}
$$

3. Equate the two sides:
$$
\frac{2x + b - a}{a} = \frac{4bx - 2b^2}{a^2}
$$

4. Eliminate the denominators by multiplying through by \(a^2\):
$$
a(2x + b - a) = 4bx - 2b^2
$$
$$
2ax + ab - a^2 = 4bx - 2b^2
$$

5. Combine like terms:
$$
2ax - 4bx = -2b^2 - ab + a^2
$$
$$
x(2a - 4b) = a^2 - ab - 2b^2
$$

6. Solve for \(x\):
$$
x = \frac{a^2 - ab - 2b^2}{2a - 4b}
$$
$$
x = \frac{a^2 - ab - 2b^2}{2(a - 2b)}
$$

Answer:
$$
\boxed{\frac{a^2 - ab - 2b^2}{2(a - 2b)}}
$$

---

#### 18. $\frac{5}{6y} + \frac{2}{3y} = 1 + \frac{1}{2y}$

1. Eliminate the fractions by finding a common denominator (6y):
$$
6y \left( \frac{5}{6y} + \frac{2}{3y} \right) = 6y \left( 1 + \frac{1}{2y} \right)
$$
$$
6y \cdot \frac{5}{6y} + 6y \cdot \frac{2}{3y} = 6y \cdot 1 + 6y \cdot \frac{1}{2y}
$$
$$
5 + 2 \cdot 2 = 6y + 3
$$
$$
5 + 4 = 6y + 3
$$
$$
9 = 6y + 3
$$

2. Combine like terms:
$$
6y = 9 - 3
$$
$$
6y = 6
$$

3. Solve for \(y\):
$$
y = \frac{6}{6} = 1
$$

Answer:
$$
\boxed{1}
$$

---

Final Answers:

1. $\boxed{12}$
2. $\boxed{12}$
3. $\boxed{-\frac{3}{7}}$
4. $\boxed{12}$
5. $\boxed{2}$
6. $\boxed{5}$
7. $\boxed{4}$
8. $\boxed{8}$
9. $\boxed{-\frac{3}{2}}$
10. $\boxed{8}$
11. $\boxed{1}$
12. $\boxed{-1}$
13. $\boxed{4}$
14. $\boxed{\frac{19}{7}}$
15. $\boxed{\frac{13}{20}}$
16. $\boxed{4}$
17. $\boxed{\frac{a^2 - ab - 2b^2}{2(a - 2b)}}$
18. $\boxed{1}$