Multiplying fractions within linear equations can seem daunting at first, but with the right techniques and a bit of practice, it becomes straightforward. This guide breaks down the process into easy-to-understand steps, ensuring you master this essential algebra skill.
Understanding the Fundamentals
Before tackling multiplication of fractions in linear equations, let's refresh some fundamental concepts:
What are Linear Equations?
Linear equations are algebraic equations where the highest power of the variable (usually 'x' or 'y') is 1. They represent a straight line when graphed. Examples include:
- 2x + 5 = 9
- y - 3 = 4x
- 1/2x = 7
Multiplying Fractions: A Quick Refresher
Remember the basic rule: Multiply the numerators (top numbers) together, and multiply the denominators (bottom numbers) together.
- (a/b) * (c/d) = (ac) / (bd)
For example: (2/3) * (4/5) = (24) / (35) = 8/15
Multiplying Fractions in Linear Equations: Step-by-Step
Let's illustrate with examples:
Example 1: Solving for x
(1/2)x + 3 = 7
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Isolate the term with the fraction: Subtract 3 from both sides:
(1/2)x = 4
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Eliminate the fraction: To get rid of the fraction (1/2), multiply both sides of the equation by its reciprocal (2/1 or simply 2):
2 * (1/2)x = 4 * 2
x = 8
Example 2: More Complex Fractions
(3/4)x - (2/5) = 1
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Isolate the terms with fractions: Add (2/5) to both sides:
(3/4)x = 1 + (2/5)
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Find a common denominator (if needed): To add 1 and (2/5), rewrite 1 as (5/5):
(3/4)x = (5/5) + (2/5) = (7/5)
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Eliminate the fraction: Multiply both sides by the reciprocal of (3/4), which is (4/3):
(4/3) * (3/4)x = (7/5) * (4/3)
x = 28/15
Example 3: Fractions on Both Sides
(2/3)x + 1 = (1/2)x + 5
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Gather x terms: Subtract (1/2)x from both sides:
(2/3)x - (1/2)x + 1 = 5
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Find a common denominator: The common denominator for (2/3) and (1/2) is 6:
(4/6)x - (3/6)x + 1 = 5
(1/6)x + 1 = 5
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Isolate the x term: Subtract 1 from both sides:
(1/6)x = 4
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Solve for x: Multiply both sides by 6:
x = 24
Tips for Success
- Practice regularly: The more you practice, the more comfortable you'll become.
- Break down problems: Tackle each step methodically.
- Check your work: Substitute your answer back into the original equation to ensure it's correct.
- Use online resources: Numerous websites and videos offer further explanations and practice problems.
Mastering fraction multiplication in linear equations is a crucial stepping stone in algebra. By following these techniques and dedicating time to practice, you'll confidently solve even the most challenging equations. Remember, consistent effort is key to success!
