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Chapter 5: Problem 17

Solve proportion. \(\frac{a+2}{a}=\frac{7}{5}\)

Short Answer

Expert verified
The solution to the proportion is \(a = 5\).

Step by step solution

01

Understand the Proportion

We start with the proportion equation \(\frac{a+2}{a}=\frac{7}{5}\). Our goal is to find the value of \(a\) that satisfies this equation.
02

Cross-Multiply

To eliminate the fractions, cross-multiply the terms: \[ (a + 2) \times 5 = a \times 7 \]
03

Expand and Simplify

Distribute the multiplication on each side: \[ 5(a + 2) = 7a \Rightarrow 5a + 10 = 7a \]
04

Isolate the Variable

Move all terms involving \(a\) to one side of the equation: \[ 5a + 10 - 5a = 7a - 5a \Rightarrow 10 = 2a \]
05

Solve for \(a\)

Divide both sides of the equation by 2: \[ \frac{10}{2} = a \Rightarrow a = 5 \]

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Key Concepts

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

cross-multiplication
Cross-multiplication is a technique used to solve proportions. When you have a proportion in the form of \( \frac{a}{b} = \frac{c}{d} \), you can eliminate the fractions by cross-multiplying. This means multiplying the numerator of one fraction by the denominator of the other fraction. For example, given the proportion \( \frac{a+2}{a} = \frac{7}{5} \), we cross-multiply to get
  • \( (a + 2) \times 5 = a \times 7 \)
This step helps to convert a proportion into a linear equation, making it easier to solve.
equation solving
Once we have cross-multiplied, we end up with an equation that needs to be solved. For instance:
  • \( (a + 2) \times 5 = 7a \)
Solving this equation involves simplifying both sides to find the value of the variable. Starting from here, we distribute and combine like terms. This is a crucial part in equation solving, ensuring that we isolate the variable efficiently.
variable isolation
Isolating the variable means getting the variable (in this case, \(a\)) all by itself on one side of the equation. After we have distributed and formed our equation:
  • \( 5a + 10 = 7a \)
We move terms involving \(a\) to one side:
  • \( 5a + 10 - 5a = 7a - 5a \)
This simplifies to:
  • \( 10 = 2a \)
Finally, we divide both sides by 2 to find the value of \(a\). This process of isolating the variable is key to solving linear equations.
distributive property
The distributive property helps us to expand expressions and simplify equations. When you see an expression like \( 5(a + 2) \), distribute the 5 to both \(a\) and 2:
  • \( 5(a + 2) = 5a + 10 \)
Similarly, applying the distributive property correctly ensures that both sides of the equation are simplified appropriately. Once we do this, we can easily proceed with isolating the variable and solving the equation.

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