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Chapter 5: Problem 18

Solve proportion. \(\frac{x}{x-3}=\frac{11}{8}\)

Short Answer

Expert verified
x = 11

Step by step solution

01

Write the proportion

The given proportion is \(\frac{x}{x-3}=\frac{11}{8}\).
02

Cross multiply

To solve the proportion, cross multiply to get a single equation: \(\frac{x}{x-3} \times 8 = 11 \times \frac{8}{8}\) becomes \(8x = 11(x-3)\).
03

Distribute

Distribute 11 on the right side: \(8x = 11x - 33\).
04

Isolate x

Move all terms involving x to one side of the equation by subtracting 11x from both sides: \(8x - 11x = -33\) simplifies to \(-3x = -33\).
05

Solve for x

Divide both sides by -3 to solve for x: \(\frac{-33}{-3} = x\), which simplifies to \(x = 11\).

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Key Concepts

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

Concept of Cross Multiplication in Proportions
When solving proportions, an essential method used is cross multiplication. This technique helps to eliminate the fraction, making the equation easier to work with. Let's examine the given problem: \( \frac{x}{x-3} = \frac{11}{8} \). Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction: \[ \frac{x}{x-3} \times 8 = 11 \times \frac{8}{8} \] This can be simplified to: \[ 8x = 11(x-3) \] Cross multiplication turns the proportion into a standard algebraic equation. This simplifies the problem-solving process.
Isolating the Variable in an Equation
After cross multiplication, our new equation is \( 8x = 11(x-3) \). The next critical step is isolating the variable, which in this case is \( x \). We want all \( x \) terms on one side of the equation: \[ 8x = 11x - 33 \] To isolate \( x \), subtract 11x from both sides: \[ 8x - 11x = -33 \] This simplifies to: \[ -3x = -33 \] Finally, divide both sides by -3 to solve for \( x \): \[ \frac{-33}{-3} = x \] Which simplifies to: \[ x = 11 \] Isolating the variable means rearranging and simplifying the equation so that \( x \) stands alone on one side.
Using the Distributive Property
In the process of solving equations, we frequently use the distributive property. The distributive property states that: \[ a(b + c) = ab + ac \] This property helps in simplifying the equation. In the given problem, we've cross multiplied to get: \[ 8x = 11(x - 3) \] Applying the distributive property on the right-hand side, we have: \[ 8x = 11x - 33 \] Here, 11 is distributed through the parentheses to both \( x \) and \( -3 \). This step is crucial as it helps break down the equation into simpler terms, paving the way for isolating the variable and ultimately solving for \( x \).

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