Solving Equations with Fractions Worksheets - Math Monks - Free Printable

Problem: Solve the multi-step equations with fractions.

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

---

#### Equation 1:
\[
\frac{x}{3} - \frac{x-2}{2} = \frac{13}{5}
\]

1. Find a common denominator for the fractions on the left-hand side. The denominators are 3 and 2, so the least common denominator (LCD) is 6.
2. Rewrite each fraction with the LCD:
\[
\frac{x}{3} = \frac{2x}{6}, \quad \frac{x-2}{2} = \frac{3(x-2)}{6}
\]
3. Substitute these into the equation:
\[
\frac{2x}{6} - \frac{3(x-2)}{6} = \frac{13}{5}
\]
4. Combine the fractions on the left-hand side:
\[
\frac{2x - 3(x-2)}{6} = \frac{13}{5}
\]
5. Simplify the numerator:
\[
2x - 3(x-2) = 2x - 3x + 6 = -x + 6
\]
So the equation becomes:
\[
\frac{-x + 6}{6} = \frac{13}{5}
\]
6. Eliminate the denominators by cross-multiplying:
\[
5(-x + 6) = 13 \cdot 6
\]
\[
-5x + 30 = 78
\]
7. Solve for \( x \):
\[
-5x = 78 - 30
\]
\[
-5x = 48
\]
\[
x = -\frac{48}{5}
\]

Solution for Equation 1:
\[
\boxed{-\frac{48}{5}}
\]

---

#### Equation 2:
\[
\frac{x-3}{6} = \frac{x-2}{4} - 36
\]

1. Find a common denominator for the fractions. The denominators are 6 and 4, so the LCD is 12.
2. Rewrite each fraction with the LCD:
\[
\frac{x-3}{6} = \frac{2(x-3)}{12}, \quad \frac{x-2}{4} = \frac{3(x-2)}{12}
\]
3. Substitute these into the equation:
\[
\frac{2(x-3)}{12} = \frac{3(x-2)}{12} - 36
\]
4. Eliminate the denominators by multiplying through by 12:
\[
2(x-3) = 3(x-2) - 432
\]
5. Distribute and simplify:
\[
2x - 6 = 3x - 6 - 432
\]
\[
2x - 6 = 3x - 438
\]
6. Solve for \( x \):
\[
2x - 3x = -438 + 6
\]
\[
-x = -432
\]
\[
x = 432
\]

Solution for Equation 2:
\[
\boxed{432}
\]

---

#### Equation 3:
\[
\frac{2x-1}{3} + \frac{x+2}{4} = \frac{8}{9}
\]

1. Find a common denominator for the fractions. The denominators are 3, 4, and 9, so the LCD is 36.
2. Rewrite each fraction with the LCD:
\[
\frac{2x-1}{3} = \frac{12(2x-1)}{36}, \quad \frac{x+2}{4} = \frac{9(x+2)}{36}, \quad \frac{8}{9} = \frac{32}{36}
\]
3. Substitute these into the equation:
\[
\frac{12(2x-1)}{36} + \frac{9(x+2)}{36} = \frac{32}{36}
\]
4. Combine the fractions on the left-hand side:
\[
\frac{12(2x-1) + 9(x+2)}{36} = \frac{32}{36}
\]
5. Simplify the numerator:
\[
12(2x-1) + 9(x+2) = 24x - 12 + 9x + 18 = 33x + 6
\]
So the equation becomes:
\[
\frac{33x + 6}{36} = \frac{32}{36}
\]
6. Eliminate the denominators by cross-multiplying:
\[
33x + 6 = 32
\]
7. Solve for \( x \):
\[
33x = 32 - 6
\]
\[
33x = 26
\]
\[
x = \frac{26}{33}
\]

Solution for Equation 3:
\[
\boxed{\frac{26}{33}}
\]

---

#### Equation 4:
\[
2 - \frac{1-x}{27} = \frac{5x+3}{12}
\]

1. Eliminate the fractions by finding a common denominator. The denominators are 27 and 12, so the LCD is 108.
2. Rewrite each fraction with the LCD:
\[
2 = \frac{216}{108}, \quad \frac{1-x}{27} = \frac{4(1-x)}{108}, \quad \frac{5x+3}{12} = \frac{9(5x+3)}{108}
\]
3. Substitute these into the equation:
\[
\frac{216}{108} - \frac{4(1-x)}{108} = \frac{9(5x+3)}{108}
\]
4. Combine the fractions on the left-hand side:
\[
\frac{216 - 4(1-x)}{108} = \frac{9(5x+3)}{108}
\]
5. Simplify the numerators:
\[
216 - 4(1-x) = 216 - 4 + 4x = 212 + 4x
\]
\[
9(5x+3) = 45x + 27
\]
So the equation becomes:
\[
\frac{212 + 4x}{108} = \frac{45x + 27}{108}
\]
6. Eliminate the denominators by cross-multiplying:
\[
212 + 4x = 45x + 27
\]
7. Solve for \( x \):
\[
212 - 27 = 45x - 4x
\]
\[
185 = 41x
\]
\[
x = \frac{185}{41}
\]

Solution for Equation 4:
\[
\boxed{\frac{185}{41}}
\]

---

#### Equation 5:
\[
8 - \frac{9-x}{24} = \frac{8x+31}{9}
\]

1. Eliminate the fractions by finding a common denominator. The denominators are 24 and 9, so the LCD is 72.
2. Rewrite each fraction with the LCD:
\[
8 = \frac{576}{72}, \quad \frac{9-x}{24} = \frac{3(9-x)}{72}, \quad \frac{8x+31}{9} = \frac{8(8x+31)}{72}
\]
3. Substitute these into the equation:
\[
\frac{576}{72} - \frac{3(9-x)}{72} = \frac{8(8x+31)}{72}
\]
4. Combine the fractions on the left-hand side:
\[
\frac{576 - 3(9-x)}{72} = \frac{8(8x+31)}{72}
\]
5. Simplify the numerators:
\[
576 - 3(9-x) = 576 - 27 + 3x = 549 + 3x
\]
\[
8(8x+31) = 64x + 248
\]
So the equation becomes:
\[
\frac{549 + 3x}{72} = \frac{64x + 248}{72}
\]
6. Eliminate the denominators by cross-multiplying:
\[
549 + 3x = 64x + 248
\]
7. Solve for \( x \):
\[
549 - 248 = 64x - 3x
\]
\[
301 = 61x
\]
\[
x = \frac{301}{61}
\]

Solution for Equation 5:
\[
\boxed{\frac{301}{61}}
\]

---

#### Equation 6:
\[
\frac{9(x+3)}{2} - 6x = 3(4-2x)
\]

1. Distribute the terms:
\[
\frac{9(x+3)}{2} = \frac{9x + 27}{2}
\]
\[
3(4-2x) = 12 - 6x
\]
So the equation becomes:
\[
\frac{9x + 27}{2} - 6x = 12 - 6x
\]
2. Eliminate the fraction by multiplying through by 2:
\[
9x + 27 - 12x = 24 - 12x
\]
3. Simplify:
\[
-3x + 27 = 24 - 12x
\]
4. Solve for \( x \):
\[
-3x + 12x = 24 - 27
\]
\[
9x = -3
\]
\[
x = -\frac{1}{3}
\]

Solution for Equation 6:
\[
\boxed{-\frac{1}{3}}
\]

---

#### Equation 7:
\[
\frac{3}{4}x + \frac{1}{2} = \frac{8}{5}x + \frac{9}{8}
\]

1. Eliminate the fractions by finding a common denominator. The denominators are 4, 2, 5, and 8, so the LCD is 40.
2. Rewrite each term with the LCD:
\[
\frac{3}{4}x = \frac{30x}{40}, \quad \frac{1}{2} = \frac{20}{40}, \quad \frac{8}{5}x = \frac{64x}{40}, \quad \frac{9}{8} = \frac{45}{40}
\]
3. Substitute these into the equation:
\[
\frac{30x}{40} + \frac{20}{40} = \frac{64x}{40} + \frac{45}{40}
\]
4. Combine the fractions:
\[
\frac{30x + 20}{40} = \frac{64x + 45}{40}
\]
5. Eliminate the denominators by cross-multiplying:
\[
30x + 20 = 64x + 45
\]
6. Solve for \( x \):
\[
30x - 64x = 45 - 20
\]
\[
-34x = 25
\]
\[
x = -\frac{25}{34}
\]

Solution for Equation 7:
\[
\boxed{-\frac{25}{34}}
\]

---

#### Equation 8:
\[
\frac{2}{3}(x-7) = \frac{16}{19}(x+4)
\]

1. Distribute the terms:
\[
\frac{2}{3}(x-7) = \frac{2x - 14}{3}
\]
\[
\frac{16}{19}(x+4) = \frac{16x + 64}{19}
\]
So the equation becomes:
\[
\frac{2x - 14}{3} = \frac{16x + 64}{19}
\]
2. Eliminate the fractions by cross-multiplying:
\[
19(2x - 14) = 3(16x + 64)
\]
3. Distribute and simplify:
\[
38x - 266 = 48x + 192
\]
4. Solve for \( x \):
\[
38x - 48x = 192 + 266
\]
\[
-10x = 458
\]
\[
x = -\frac{458}{10} = -\frac{229}{5}
\]

Solution for Equation 8:
\[
\boxed{-\frac{229}{5}}
\]

---

#### Equation 9:
\[
\frac{3-2x}{5} - \frac{4x-3}{4} = -9
\]

1. Find a common denominator for the fractions. The denominators are 5 and 4, so the LCD is 20.
2. Rewrite each fraction with the LCD:
\[
\frac{3-2x}{5} = \frac{4(3-2x)}{20}, \quad \frac{4x-3}{4} = \frac{5(4x-3)}{20}
\]
3. Substitute these into the equation:
\[
\frac{4(3-2x)}{20} - \frac{5(4x-3)}{20} = -9
\]
4. Combine the fractions on the left-hand side:
\[
\frac{4(3-2x) - 5(4x-3)}{20} = -9
\]
5. Simplify the numerator:
\[
4(3-2x) - 5(4x-3) = 12 - 8x - 20x + 15 = 27 - 28x
\]
So the equation becomes:
\[
\frac{27 - 28x}{20} = -9
\]
6. Eliminate the denominators by cross-multiplying:
\[
27 - 28x = -180
\]
7. Solve for \( x \):
\[
27 + 180 = 28x
\]
\[
207 = 28x
\]
\[
x = \frac{207}{28}
\]

Solution for Equation 9:
\[
\boxed{\frac{207}{28}}
\]

---

#### Equation 10:
\[
\frac{6-5x}{15} - \frac{7x+9}{7} = 14
\]

1. Find a common denominator for the fractions. The denominators are 15 and 7, so the LCD is 105.
2. Rewrite each fraction with the LCD:
\[
\frac{6-5x}{15} = \frac{7(6-5x)}{105}, \quad \frac{7x+9}{7} = \frac{15(7x+9)}{105}
\]
3. Substitute these into the equation:
\[
\frac{7(6-5x)}{105} - \frac{15(7x+9)}{105} = 14
\]
4. Combine the fractions on the left-hand side:
\[
\frac{7(6-5x) - 15(7x+9)}{105} = 14
\]
5. Simplify the numerator:
\[
7(6-5x) - 15(7x+9) = 42 - 35x - 105x - 135 = -140x - 93
\]
So the equation becomes:
\[
\frac{-140x - 93}{105} = 14
\]
6. Eliminate the denominators by cross-multiplying:
\[
-140x - 93 = 14 \cdot 105
\]
\[
-140x - 93 = 1470
\]
7. Solve for \( x \):
\[
-140x = 1470 + 93
\]
\[
-140x = 1563
\]
\[
x = -\frac{1563}{140}
\]

Solution for Equation 10:
\[
\boxed{-\frac{1563}{140}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -\frac{48}{5} \\
2. & \ 432 \\
3. & \ \frac{26}{33} \\
4. & \ \frac{185}{41} \\
5. & \ \frac{301}{61} \\
6. & \ -\frac{1}{3} \\
7. & \ -\frac{25}{34} \\
8. & \ -\frac{229}{5} \\
9. & \ \frac{207}{28} \\
10. & \ -\frac{1563}{140}
\end{aligned}
}
\]