Let’s solve each example step by step.
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Ex 1:
Equation:
\[\frac{x - 7}{5} = 7\]
Step 1: Multiply both sides by the denominator (5) to clear the fraction.
\[5 \cdot \frac{x - 7}{5} = 5 \cdot 7\]
→ \(x - 7 = 35\)
Step 2: Add 7 to both sides to solve for x.
\[x - 7 + 7 = 35 + 7\]
→ \(x = 42\)
✔ Check: Plug back in: \(\frac{42 - 7}{5} = \frac{35}{5} = 7\) → Correct!
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Ex 2:
Equation:
\[\frac{1}{3}(x - 6) + 4 = x - 6\]
Step 1: The only fraction is \(\frac{1}{3}\), so multiply EVERY term on BOTH sides by 3.
\[3 \cdot \left[ \frac{1}{3}(x - 6) + 4 \right] = 3 \cdot (x - 6)\]
Distribute the 3:
→ \(1 \cdot (x - 6) + 12 = 3(x - 6)\)
→ \(x - 6 + 12 = 3x - 18\)
Simplify left side:
→ \(x + 6 = 3x - 18\)
Step 2: Get all x terms on one side, constants on the other.
Subtract x from both sides:
→ \(6 = 2x - 18\)
Add 18 to both sides:
→ \(24 = 2x\)
Divide by 2:
→ \(x = 12\)
✔ Check: Left side: \(\frac{1}{3}(12 - 6) + 4 = \frac{6}{3} + 4 = 2 + 4 = 6\)
Right side: \(12 - 6 = 6\) → Correct!
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Ex 3:
Equation:
\[\frac{2}{3}x + 1 = -5 + \frac{5}{6}x\]
Step 1: Denominators are 3 and 6 → LCM is 6. Multiply EVERY term by 6.
\[6 \cdot \left( \frac{2}{3}x + 1 \right) = 6 \cdot \left( -5 + \frac{5}{6}x \right)\]
Distribute:
→ \(6 \cdot \frac{2}{3}x + 6 \cdot 1 = 6 \cdot (-5) + 6 \cdot \frac{5}{6}x\)
→ \(4x + 6 = -30 + 5x\)
Step 2: Move variable terms to one side, constants to the other.
Subtract 4x from both sides:
→ \(6 = -30 + x\)
Add 30 to both sides:
→ \(36 = x\)
So, \(x = 36\)
✔ Check:
Left: \(\frac{2}{3}(36) + 1 = 24 + 1 = 25\)
Right: \(-5 + \frac{5}{6}(36) = -5 + 30 = 25\) → Correct!
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Final Answer:
Ex 1: x = 42
Ex 2: x = 12
Ex 3: x = 36
Parent Tip: Review the logic above to help your child master the concept of clearing fractions worksheet.
