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Chapter 6: Problem 37

Solve each equation, and check your solutions. \(\frac{x}{2}=\frac{5}{4}+\frac{x-1}{4}\)

Short Answer

Expert verified
x = 4

Step by step solution

01

- Eliminate the fractions

Multiply every term by 4 to eliminate the fractions in the equation \[ 4 \times \frac{x}{2} = 4 \times \frac{5}{4} + 4 \times \frac{x-1}{4} \]This simplifies to: \[ 2x = 5 + (x-1) \]
02

- Simplify the equation

Distribute and simplify the right side of the equation \[ 2x = 5 + x - 1 \]Combine like terms: \[ 2x = x + 4 \]
03

- Isolate the variable x

Subtract x from both sides to isolate the variable: \[ 2x - x = 4 \]This simplifies to: \[ x = 4 \]
04

- Check the solution

Substitute \( x = 4 \) back into the original equation to verify: \[ \frac{4}{2} = \frac{5}{4} + \frac{4-1}{4} \]Simplify both sides: \[ 2 = \frac{5}{4} + \frac{3}{4} \]\[ 2 = 2 \]Since both sides are equal, \( x = 4 \) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eliminating Fractions
When confronted with fractions in an equation, it can simplify the solving process to eliminate them.
Multiplying every term by the least common denominator (LCD) is an effective method to get rid of fractions.
In our example, we multiplied every term by 4, the LCD of the denominators 2 and 4, and got:

\[ 4 \times \frac{x}{2} = 4 \times \frac{5}{4} + 4 \times \frac{x-1}{4} \]

By doing this, we transformed the original equation into one without fractions:
\[ 2x = 5 + (x-1) \]

Without the fractions, the equation becomes much simpler to solve, streamlining the path towards isolating the variable.
Isolating Variables
The next step involves isolating the variable we'd like to solve for, making it the sole subject on one side of the equation.
After our equation was simplified to \[ 2x = x + 4 \], we needed to gather all \(x\) terms on one side.We achieved this by subtracting \(x\) from both sides:

\[ 2x - x = 4 \]

This simplifies directly to \(x = 4\), isolating \(x\).Isolating the variable means you can clearly see the value of \(x\) and is crucial in solving any algebraic equation.
Subtracting, adding, dividing, or multiplying both sides of the equation systematically isolates the variable.
Checking Solutions
Checking the solution is always an important final step.This ensures your solution is correct, and the original equation holds true when the solution is substituted back in.
In our example, after finding \(x = 4\), we substitute \(4\) back into the original equation:

\[ \frac{4}{2} = \frac{5}{4} + \frac{4-1}{4} \]

Simplifying both sides confirms the correctness:
\[ 2 = \frac{5}{4} + \frac{3}{4} \]
\[ 2 = 2 \]

Since both sides are equal, our solution \(x = 4\) is verified.
Checking your work can catch errors and reinforces understanding of the problem-solving process.

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