Two-Step Equations: Fractions Worksheets | Grade1to6.com - Free Printable

Let's solve each of the two-step equations involving fractions step by step. The goal is to isolate the variable $ x $ using algebraic operations.

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a. $ 7x = \frac{5}{3} - 7 $

Step 1: Simplify the right-hand side:
$$
\frac{5}{3} - 7 = \frac{5}{3} - \frac{21}{3} = \frac{-16}{3}
$$

So,
$$
7x = \frac{-16}{3}
$$

Step 2: Divide both sides by 7:
$$
x = \frac{-16}{3} \div 7 = \frac{-16}{3} \times \frac{1}{7} = \frac{-16}{21}
$$

✔ Answer: $ x = -\frac{16}{21} $

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b. $ \frac{1}{5} = 7x - 6 $

Step 1: Add 6 to both sides:
$$
\frac{1}{5} + 6 = 7x \Rightarrow \frac{1}{5} + \frac{30}{5} = \frac{31}{5}
$$

So,
$$
7x = \frac{31}{5}
$$

Step 2: Divide both sides by 7:
$$
x = \frac{31}{5} \div 7 = \frac{31}{5} \times \frac{1}{7} = \frac{31}{35}
$$

✔ Answer: $ x = \frac{31}{35} $

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c. $ -\frac{3}{3} + \frac{8}{3} = -2x $

Step 1: Simplify left side:
$$
-\frac{3}{3} + \frac{8}{3} = \frac{5}{3}
$$

So,
$$
\frac{5}{3} = -2x
$$

Step 2: Divide both sides by -2:
$$
x = \frac{5}{3} \div (-2) = \frac{5}{3} \times \left(-\frac{1}{2}\right) = -\frac{5}{6}
$$

✔ Answer: $ x = -\frac{5}{6} $

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d. $ -5x = \frac{1}{4} + 3 $

Step 1: Simplify right side:
$$
\frac{1}{4} + 3 = \frac{1}{4} + \frac{12}{4} = \frac{13}{4}
$$

So,
$$
-5x = \frac{13}{4}
$$

Step 2: Divide both sides by -5:
$$
x = \frac{13}{4} \div (-5) = \frac{13}{4} \times \left(-\frac{1}{5}\right) = -\frac{13}{20}
$$

✔ Answer: $ x = -\frac{13}{20} $

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e. $ -7x = \frac{9}{7} + 6 $

Step 1: Simplify right side:
$$
\frac{9}{7} + 6 = \frac{9}{7} + \frac{42}{7} = \frac{51}{7}
$$

So,
$$
-7x = \frac{51}{7}
$$

Step 2: Divide both sides by -7:
$$
x = \frac{51}{7} \div (-7) = \frac{51}{7} \times \left(-\frac{1}{7}\right) = -\frac{51}{49}
$$

✔ Answer: $ x = -\frac{51}{49} $

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f. $ -6x = -\frac{8}{5} + 3 $

Step 1: Simplify right side:
$$
-\frac{8}{5} + 3 = -\frac{8}{5} + \frac{15}{5} = \frac{7}{5}
$$

So,
$$
-6x = \frac{7}{5}
$$

Step 2: Divide both sides by -6:
$$
x = \frac{7}{5} \div (-6) = \frac{7}{5} \times \left(-\frac{1}{6}\right) = -\frac{7}{30}
$$

✔ Answer: $ x = -\frac{7}{30} $

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g. $ -\frac{8}{6} + \frac{9}{5} = -8x $

Step 1: Simplify left side (simplify and find common denominator):
$$
-\frac{8}{6} = -\frac{4}{3}, \quad \text{so } -\frac{4}{3} + \frac{9}{5}
$$

LCM of 3 and 5 is 15:
$$
-\frac{4}{3} = -\frac{20}{15}, \quad \frac{9}{5} = \frac{27}{15}
\Rightarrow -\frac{20}{15} + \frac{27}{15} = \frac{7}{15}
$$

So,
$$
\frac{7}{15} = -8x
$$

Step 2: Divide both sides by -8:
$$
x = \frac{7}{15} \div (-8) = \frac{7}{15} \times \left(-\frac{1}{8}\right) = -\frac{7}{120}
$$

✔ Answer: $ x = -\frac{7}{120} $

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h. $ 8x = \frac{3}{3} - 7 $

Step 1: Simplify right side:
$$
\frac{3}{3} = 1, \quad 1 - 7 = -6
$$

So,
$$
8x = -6
$$

Step 2: Divide by 8:
$$
x = -\frac{6}{8} = -\frac{3}{4}
$$

✔ Answer: $ x = -\frac{3}{4} $

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i. $ \frac{7}{3} = 6x - 6 $

Step 1: Add 6 to both sides:
$$
\frac{7}{3} + 6 = 6x \Rightarrow \frac{7}{3} + \frac{18}{3} = \frac{25}{3}
$$

So,
$$
6x = \frac{25}{3}
$$

Step 2: Divide by 6:
$$
x = \frac{25}{3} \div 6 = \frac{25}{3} \times \frac{1}{6} = \frac{25}{18}
$$

✔ Answer: $ x = \frac{25}{18} $

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j. $ 2x = \frac{8}{10} - 10 $

Step 1: Simplify right side:
$$
\frac{8}{10} = \frac{4}{5}, \quad \frac{4}{5} - 10 = \frac{4}{5} - \frac{50}{5} = -\frac{46}{5}
$$

So,
$$
2x = -\frac{46}{5}
$$

Step 2: Divide by 2:
$$
x = -\frac{46}{5} \div 2 = -\frac{46}{5} \times \frac{1}{2} = -\frac{23}{5}
$$

✔ Answer: $ x = -\frac{23}{5} $

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k. $ -2x = -\frac{8}{6} + 6 $

Step 1: Simplify:
$$
-\frac{8}{6} = -\frac{4}{3}, \quad -\frac{4}{3} + 6 = -\frac{4}{3} + \frac{18}{3} = \frac{14}{3}
$$

So,
$$
-2x = \frac{14}{3}
$$

Step 2: Divide by -2:
$$
x = \frac{14}{3} \div (-2) = \frac{14}{3} \times \left(-\frac{1}{2}\right) = -\frac{14}{6} = -\frac{7}{3}
$$

✔ Answer: $ x = -\frac{7}{3} $

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l. $ 10x = -\frac{9}{2} - 5 $

Step 1: Simplify right side:
$$
-\frac{9}{2} - 5 = -\frac{9}{2} - \frac{10}{2} = -\frac{19}{2}
$$

So,
$$
10x = -\frac{19}{2}
$$

Step 2: Divide by 10:
$$
x = -\frac{19}{2} \div 10 = -\frac{19}{2} \times \frac{1}{10} = -\frac{19}{20}
$$

✔ Answer: $ x = -\frac{19}{20} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| a. | $ x = -\frac{16}{21} $ |
| b. | $ x = \frac{31}{35} $ |
| c. | $ x = -\frac{5}{6} $ |
| d. | $ x = -\frac{13}{20} $ |
| e. | $ x = -\frac{51}{49} $ |
| f. | $ x = -\frac{7}{30} $ |
| g. | $ x = -\frac{7}{120} $ |
| h. | $ x = -\frac{3}{4} $ |
| i. | $ x = \frac{25}{18} $ |
| j. | $ x = -\frac{23}{5} $ |
| k. | $ x = -\frac{7}{3} $ |
| l. | $ x = -\frac{19}{20} $ |

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📌 Tips for Solving These:

1. Simplify constants first, especially fractions.
2. Use common denominators when adding/subtracting fractions.
3. Isolate the variable term on one side.
4. Divide both sides by the coefficient of $ x $.
5. Always simplify final answers (reduce fractions).

Let me know if you'd like these steps explained in a video format or need practice problems!