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Chapter 6: Problem 20

Solve each equation, and check the solutions. $$ \frac{4 x+3}{6}=\frac{5}{2} $$

Short Answer

Expert verified
x = 3

Step by step solution

01

- Eliminate the Fraction

Multiply both sides of the equation by 6 to eliminate the fraction. \[\frac{4x + 3}{6} \times 6 = \frac{5}{2} \times 6\] This simplifies to: \[4x + 3 = 15\]
02

- Isolate the Variable Term

To isolate 4x, subtract 3 from both sides of the equation. \[4x + 3 - 3 = 15 - 3\] This gives us: \[4x = 12\]
03

- Solve for x

Divide both sides by 4 to solve for x. \[4x / 4 = 12 / 4\] This simplifies to: \[x = 3\]
04

- Check the Solution

Substitute x = 3 back into the original equation to verify if both sides are equal. \[\frac{4(3) + 3}{6} = \frac{5}{2}\] Simplifying inside the fraction: \[\frac{12 + 3}{6} = \frac{15}{6} = \frac{5}{2}\] Both sides are equal, hence x = 3 is correct.

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Key Concepts

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

Fraction Elimination
When solving rational equations, the first step often involves eliminating fractions. This makes the equation simpler and easier to handle. For instance, given the equation \(\frac{4x+3}{6}=\frac{5}{2}\), you can eliminate the fractions by finding a common multiple and multiplying all terms by it. Here, we multiply both sides by 6, the common denominator for the fractions involved. This simplifies the equation to standard linear form, which is easier to solve.
Isolating Variables
Isolating the variable is a crucial step in solving equations. In our case, after eliminating the fraction, we have the equation \(4x + 3 = 15\). To isolate the variable term \(4x\), subtract 3 from both sides. This leaves you with \(4x = 12\). Isolation helps in pinpointing the variable so that subsequent steps, like division, become straightforward.
Checking Solutions
Checking your solution ensures accuracy and verifies that your steps were correct. Once you solve for \(x\), substitute \(x = 3\) back into the original equation \(\frac{4(3)+3}{6} = \frac{5}{2}\). Simplify the left side of the equation to confirm it matches the right side. This step helps catch any mistakes and validates your final answer.
Equation Simplification
Simplifying equations involves making them as basic as possible. By multiplying by 6 to eliminate fractions and then isolating the variable, we've reduced our complex rational equation to a simple form. Simplification involves operations like multiplication, division, addition, and subtraction. Each step should aim to reduce the complexity while keeping the balance of the equation intact. This ensures a more manageable equation and easier solution process.

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