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Chapter 1: Problem 26

Linear Equations The given equation is either linear or equivalent to a linear equation. Solve the equation. $$\frac{2 x-1}{x+2}=\frac{4}{5}$$

Short Answer

Expert verified
The solution is \( x = \frac{13}{6} \).

Step by step solution

01

Cross-Multiply to Eliminate Fractions

The given equation is \( \frac{2x - 1}{x + 2} = \frac{4}{5} \). To eliminate the fractions, cross-multiply to obtain: \[ 5(2x - 1) = 4(x + 2) \] This changes the given equation into one that is easier to solve.
02

Expand Both Sides

Now, expand both sides of the equation:For the left side: \[ 5(2x - 1) = 5 \times 2x - 5 \times 1 = 10x - 5 \]For the right side:\[ 4(x + 2) = 4 \times x + 4 \times 2 = 4x + 8 \]
03

Combine Like Terms

We now have: \[ 10x - 5 = 4x + 8 \]Subtract \(4x\) from both sides to bring all terms with \(x\) to one side:\[ 10x - 4x - 5 = 8 \] This simplifies to:\[ 6x - 5 = 8 \]
04

Isolate the Variable Term

Add 5 to both sides to isolate the term with \(x\):\[ 6x - 5 + 5 = 8 + 5 \] This simplifies to:\[ 6x = 13 \]
05

Solve for x

Finally, divide both sides by 6 to solve for \(x\):\[ x = \frac{13}{6} \]

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Key Concepts

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

Understanding Cross-Multiplication
Cross-multiplication is a method used to eliminate fractions from an equation, making it easier to solve. To use cross-multiplication, you multiply the numerator of one fraction by the denominator of the other and set these two products equal to each other.
For the equation \( \frac{2x - 1}{x + 2} = \frac{4}{5} \), you cross-multiply to get \( 5(2x - 1) = 4(x + 2) \). This helps transform the equation with fractions into a linear equation without fractions, simplifying the path to solving for the variable.
Cross-multiplication is effective because it respects the equality of proportions, and it's often the first step when you encounter equations involving fractions.
Effective Approaches to Solving Equations
Solving equations involves finding the value of an unknown variable that makes the equation true. For linear equations, such as the one in the example, the goal is to simplify and solve step by step.
Once cross-multiplication has been done, you need to expand both sides: \( 5(2x - 1) = 10x - 5 \) and \( 4(x + 2) = 4x + 8 \).
Through expansion, each side of the equation is simplified, making it easier to manage and combine like terms.
  • This step-by-step process helps in organizing the equation, reducing complexity.
  • Moving terms systematically from one side of the equation to the other is crucial.
By simplifying the expressions, you give yourself clearer pathways to manipulate and solve the equation.
Mastering Variable Isolation
Variable isolation is the process of manipulating an equation to 'isolate' the variable on one side of the equation. The goal is to find the value of the variable by itself.
For this exercise, after simplifying the equation to \( 6x - 5 = 8 \), you add 5 to both sides, yielding \( 6x = 13 \).
The final step is to divide both sides by 6 to solve for \(x\). This process involves ensuring that the variable stands alone by eliminating other numerical components around it.
  • Adding or subtracting terms helps centralize the variable term.
  • Dividing both sides by the coefficient of the variable provides the final solution.
This step requires careful arithmetic to ensure accuracy when finding the variable's value.

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Most popular questions from this chapter

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