Problem 67 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{4 x+1}{3}=\frac{x+5}{6}+\frac{x-3}{6}\)

Step-by-Step Solution

Answer

x = 0

Step 1: Simplify the right side of the equation

Combine the fractions on the right side: \ \(\frac{4 x+1}{3}=\frac{x+5}{6}+\frac{x-3}{6}\)

Step 2: Combine the fractions on the right side

Add the fractions on the right side: \ \(\frac{4x+1}{3}=\frac{(x+5) + (x-3)}{6}\) \ Combine the terms in the numerator: \ \(\frac{4x+1}{3}=\frac{2x+2}{6}\)

Step 3: Clear the fractions by multiplying both sides by 6

Multiply both sides by 6 to eliminate the denominators: \ \(6 \times \frac{4x+1}{3} = 6 \times \frac{2x+2}{6}\) \ Simplify: \ \(2(4x+1) = 2x+2\)

Step 4: Distribute and simplify

Distribute the 2 on the left side: \ \(8x + 2 = 2x + 2\)

Step 5: Isolate the variable

Subtract 2x from both sides: \ \(8x - 2x + 2 = 2\) \ Simplify: \ \(6x + 2 = 2\) \ Subtract 2 from both sides: \ \(6x = 0\) \ Divide by 6: \ \(x = 0\)

Step 6: Check the solution

Substitute \(x = 0\) back into the original equation to verify: \ \(\frac{4(0)+1}{3}=\frac{0+5}{6}+\frac{0-3}{6}\) \ Simplify both sides: \ \(\frac{1}{3} = \frac{5}{6} + \frac{-3}{6}\) \ \(\frac{1}{3} = \frac{2}{6}\) \ \(\frac{1}{3} = \frac{1}{3}\), which is true. Therefore, \( x = 0 \) is the correct solution.

Key Concepts

Fractions

Understanding fractions is crucial when solving linear equations that include them. A fraction consists of a numerator and a denominator. For example, in \(\frac{4x+1}{3}\), \'4x+1\' is the numerator and \'3\' is the denominator.
To solve equations with fractions, you often need to combine or simplify the fractions. This involves finding a common denominator and ensuring all fractions are combined correctly. In this exercise, we had two fractions on the right side of the equation, \(\frac{x+5}{6}\) and \(\frac{x-3}{6}\).

Here are some key steps to handle fractions:

• Find a common denominator if combining fractions.
• Add or subtract the numerators while keeping the common denominator.
• Simplify if possible before proceeding.

By consistently following these steps, you can tackle more complex equations with confidence.

Variable Isolation

Isolating the variable is a critical step in solving linear equations. The goal is to get the variable (often represented by \'x\') alone on one side of the equation. This involves several possible steps including addition, subtraction, multiplication, and division.

For example, in our exercise, after combining and simplifying the fractions, we ended up with the equation \(8x + 2 = 2x + 2\). Our goal is to isolate \'x\'.

We follow these steps to achieve variable isolation:

• Subtract \2x\ from both sides to group the \'x\' terms together: \(8x - 2x + 2 = 2\) simplifies to \(6x + 2 = 2\).
• Next, subtract 2 from both sides to isolate the \'6x\' term: \(6x = 0\).
• Finally, divide by 6 to solve for \'x\': \(x = 0\).

Isolating the variable makes it easier to find the solution and ensures that you can verify the results by plugging the value back into the original equation.

Equation Simplification

Simplifying an equation is the process of making it easier to work with. This involves combining like terms, distributing constants, and eliminating fractions or complex denominators.

In our example, simplification started with the fractions on the right side of the equation. We combined \(\frac{x+5}{6} + \frac{x-3}{6}\) by adding the numerators: \(\frac{(x+5) + (x-3)}{6} = \frac{2x+2}{6}\).

After combining terms, we cleared the fractions by multiplying both sides by 6. This step removes the denominators: \(6 \times \frac{4x+1}{3} = 6 \times \frac{2x+2}{6}\) which simplifies to \(2(4x+1) = 2x+2\).

Simplifying further, we distribute the 2 on the left side, giving us \(8x + 2 = 2x + 2\). Each step brought us closer to a simpler form of the equation, making it easier to isolate \x\ and solve the problem.
Simplification helps in:

• Reducing the complexity of the equation.
• Avoiding common mistakes by working with simpler expressions.
• Speeding up the process of finding the solution.

Master these techniques, and equation simplification will become second nature.