Solving Equations with Fractions
Some equations have fractions in them, such as x/3 + 2 = 5 or (2x + 1)/4 = 3. To solve these equations, you can either clear the fractions first by multiplying both sides by the denominator (or LCM), or work with the fractions directly.
The goal is the same as any equation: isolate the variable to find its value.
What is Solving Equations with Fractions - Grade 7 Maths (Simple Equations)?
Method to solve equations with fractions:
- Identify the denominators in the equation.
- Multiply both sides by the LCM of the denominators to clear fractions.
- Solve the resulting equation as usual.
- Check the answer by substituting back.
Solving Equations with Fractions Formula
Key principle:
Whatever you do to one side, do the same to the other side.
Types and Properties
Types of fractional equations:
- x/a = b: Multiply both sides by a. x = ab.
- x/a + c = d: Subtract c, then multiply by a.
- (ax + b)/c = d: Multiply both sides by c, then solve.
- Fractions on both sides: Cross multiply.
Solved Examples
Example 1: Simple Fractional Equation
Problem: Solve x/4 = 7.
Solution:
- Multiply both sides by 4: x = 7 × 4 = 28
Check: 28/4 = 7 ✓
Answer: x = 28.
Example 2: Fraction with Addition
Problem: Solve x/3 + 5 = 9.
Solution:
- Subtract 5: x/3 = 4
- Multiply by 3: x = 12
Check: 12/3 + 5 = 4 + 5 = 9 ✓
Answer: x = 12.
Example 3: Expression in Numerator
Problem: Solve (2x + 3)/5 = 7.
Solution:
- Multiply both sides by 5: 2x + 3 = 35
- Subtract 3: 2x = 32
- Divide by 2: x = 16
Check: (2×16 + 3)/5 = 35/5 = 7 ✓
Answer: x = 16.
Example 4: Two Fractions
Problem: Solve x/2 + x/3 = 10.
Solution:
- LCM of 2 and 3 = 6
- Multiply through by 6: 3x + 2x = 60
- 5x = 60
- x = 12
Check: 12/2 + 12/3 = 6 + 4 = 10 ✓
Answer: x = 12.
Real-World Applications
Real-world uses:
- Sharing problems: “One-third of the students plus 5 equals 15. How many students?”
- Speed/time: Distance/time = speed involves fractions.
- Ratios: Many ratio problems lead to fractional equations.
Key Points to Remember
- To clear fractions, multiply both sides by the LCM of all denominators.
- Always check your answer by substituting back into the original equation.
- When the variable is in the denominator, be careful about values that make the denominator zero.
- Cross multiplication works when there is one fraction on each side.
Practice Problems
- Solve: x/5 = 8.
- Solve: y/4 − 3 = 2.
- Solve: (3x − 1)/7 = 5.
- Solve: x/2 + x/5 = 14.
Frequently Asked Questions
Q1. How do you solve an equation with fractions?
Multiply both sides of the equation by the LCM of the denominators to eliminate the fractions, then solve as usual.
Q2. What is cross multiplication?
When you have a/b = c/d, cross multiply to get ad = bc. This clears both fractions at once.
Q3. Why do we check the answer?
Checking by substitution confirms the solution is correct. It catches arithmetic errors and ensures the answer satisfies the original equation.
