Solving linear equations with two fractions - Variation Theory - Free Printable

Here are the step-by-step solutions for each equation shown in the list.

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

Step 1: Find a common denominator for 2 and 3, which is 6. Multiply every term by 6 to clear the fractions.
$$6 \cdot \left(\frac{x + 2}{2}\right) + 6 \cdot \left(\frac{x + 4}{3}\right) = 6 \cdot 4$$
$$3(x + 2) + 2(x + 4) = 24$$

Step 2: Expand the brackets.
$$3x + 6 + 2x + 8 = 24$$

Step 3: Combine like terms ($3x + 2x$ and $6 + 8$).
$$5x + 14 = 24$$

Step 4: Subtract 14 from both sides.
$$5x = 10$$

Step 5: Divide by 5.
$$x = 2$$

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2. $\frac{x - 2}{6} - \frac{x - 4}{8} = 9$

Step 1: Find the least common multiple for 6 and 8, which is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{x - 2}{6}\right) - 24 \cdot \left(\frac{x - 4}{8}\right) = 24 \cdot 9$$
$$4(x - 2) - 3(x - 4) = 216$$

Step 2: Expand the brackets. Be careful with the negative sign in front of the second bracket.
$$4x - 8 - 3x + 12 = 216$$

Step 3: Combine like terms ($4x - 3x$ and $-8 + 12$).
$$x + 4 = 216$$

Step 4: Subtract 4 from both sides.
$$x = 212$$

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

Step 1: The common denominator for 2 and 8 is 8. Multiply every term by 8.
$$8 \cdot \left(\frac{x + 2}{2}\right) + 8 \cdot \left(\frac{x + 4}{8}\right) = 8 \cdot 4$$
$$4(x + 2) + 1(x + 4) = 32$$

Step 2: Expand the brackets.
$$4x + 8 + x + 4 = 32$$

Step 3: Combine like terms.
$$5x + 12 = 32$$

Step 4: Subtract 12 from both sides.
$$5x = 20$$

Step 5: Divide by 5.
$$x = 4$$

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4. $\frac{2x - 2}{6} - \frac{2x - 4}{8} = 9$

Step 1: Simplify the fractions first if possible, or find a common denominator (24). Let's use 24.
$$24 \cdot \left(\frac{2x - 2}{6}\right) - 24 \cdot \left(\frac{2x - 4}{8}\right) = 24 \cdot 9$$
$$4(2x - 2) - 3(2x - 4) = 216$$

Step 2: Expand the brackets.
$$8x - 8 - 6x + 12 = 216$$

Step 3: Combine like terms ($8x - 6x$ and $-8 + 12$).
$$2x + 4 = 216$$

Step 4: Subtract 4 from both sides.
$$2x = 212$$

Step 5: Divide by 2.
$$x = 106$$

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5. $\frac{x + 2}{2} + \frac{x + 4}{8} = 9$

Step 1: Common denominator is 8. Multiply every term by 8.
$$8 \cdot \left(\frac{x + 2}{2}\right) + 8 \cdot \left(\frac{x + 4}{8}\right) = 8 \cdot 9$$
$$4(x + 2) + (x + 4) = 72$$

Step 2: Expand the brackets.
$$4x + 8 + x + 4 = 72$$

Step 3: Combine like terms.
$$5x + 12 = 72$$

Step 4: Subtract 12 from both sides.
$$5x = 60$$

Step 5: Divide by 5.
$$x = 12$$

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6. $\frac{4x - 2}{6} - \frac{4x - 4}{8} = 9$

Step 1: Common denominator for 6 and 8 is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{4x - 2}{6}\right) - 24 \cdot \left(\frac{4x - 4}{8}\right) = 24 \cdot 9$$
$$4(4x - 2) - 3(4x - 4) = 216$$

Step 2: Expand the brackets.
$$16x - 8 - 12x + 12 = 216$$

Step 3: Combine like terms ($16x - 12x$ and $-8 + 12$).
$$4x + 4 = 216$$

Step 4: Subtract 4 from both sides.
$$4x = 212$$

Step 5: Divide by 4.
$$x = 53$$

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7. $\frac{x + 2}{6} + \frac{x + 4}{8} = 9$

Step 1: Common denominator is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{x + 2}{6}\right) + 24 \cdot \left(\frac{x + 4}{8}\right) = 24 \cdot 9$$
$$4(x + 2) + 3(x + 4) = 216$$

Step 2: Expand the brackets.
$$4x + 8 + 3x + 12 = 216$$

Step 3: Combine like terms.
$$7x + 20 = 216$$

Step 4: Subtract 20 from both sides.
$$7x = 196$$

Step 5: Divide by 7.
$$x = 28$$

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8. $\frac{4x - 4}{8} - \frac{4x - 2}{6} = 9$

Step 1: Common denominator is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{4x - 4}{8}\right) - 24 \cdot \left(\frac{4x - 2}{6}\right) = 24 \cdot 9$$
$$3(4x - 4) - 4(4x - 2) = 216$$

Step 2: Expand the brackets.
$$12x - 12 - 16x + 8 = 216$$

Step 3: Combine like terms ($12x - 16x$ and $-12 + 8$).
$$-4x - 4 = 216$$

Step 4: Add 4 to both sides.
$$-4x = 220$$

Step 5: Divide by -4.
$$x = -55$$

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9. $\frac{x + 2}{6} - \frac{x + 4}{8} = 9$

Step 1: Common denominator is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{x + 2}{6}\right) - 24 \cdot \left(\frac{x + 4}{8}\right) = 24 \cdot 9$$
$$4(x + 2) - 3(x + 4) = 216$$

Step 2: Expand the brackets.
$$4x + 8 - 3x - 12 = 216$$

Step 3: Combine like terms.
$$x - 4 = 216$$

Step 4: Add 4 to both sides.
$$x = 220$$

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10. $\frac{4x - 4}{8} - \frac{4x - 2}{6} = 1$

Step 1: Common denominator is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{4x - 4}{8}\right) - 24 \cdot \left(\frac{4x - 2}{6}\right) = 24 \cdot 1$$
$$3(4x - 4) - 4(4x - 2) = 24$$

Step 2: Expand the brackets.
$$12x - 12 - 16x + 8 = 24$$

Step 3: Combine like terms.
$$-4x - 4 = 24$$

Step 4: Add 4 to both sides.
$$-4x = 28$$

Step 5: Divide by -4.
$$x = -7$$

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11. $\frac{x + 2}{6} - \frac{x - 4}{8} = 9$

Step 1: Common denominator is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{x + 2}{6}\right) - 24 \cdot \left(\frac{x - 4}{8}\right) = 24 \cdot 9$$
$$4(x + 2) - 3(x - 4) = 216$$

Step 2: Expand the brackets.
$$4x + 8 - 3x + 12 = 216$$

Step 3: Combine like terms.
$$x + 20 = 216$$

Step 4: Subtract 20 from both sides.
$$x = 196$$

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12. $\frac{4x - 4}{8} - \frac{4x - 2}{6} = \frac{1}{2}$

Step 1: Common denominator for 8, 6, and 2 is 24. Multiply every term by 24.
$$24 \cdot \left(\frac{4x - 4}{8}\right) - 24 \cdot \left(\frac{4x - 2}{6}\right) = 24 \cdot \left(\frac{1}{2}\right)$$
$$3(4x - 4) - 4(4x - 2) = 12$$

Step 2: Expand the brackets.
$$12x - 12 - 16x + 8 = 12$$

Step 3: Combine like terms.
$$-4x - 4 = 12$$

Step 4: Add 4 to both sides.
$$-4x = 16$$

Step 5: Divide by -4.
$$x = -4$$

Final Answer:
1. x = 2
2. x = 212
3. x = 4
4. x = 106
5. x = 12
6. x = 53
7. x = 28
8. x = -55
9. x = 220
10. x = -7
11. x = 196
12. x = -4