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Chapter 6: Problem 271

In the following exercises, solve each proportion. $$ \frac{72}{156}=\frac{-6}{q} $$

Short Answer

Expert verified
q = -13

Step by step solution

01

Set up the equation

Use the given proportion \[ \frac{72}{156}=\frac{-6}{q} \] as an equation.
02

Cross-multiply

To solve for q, cross-multiply the terms: \[ 72 \times q = 156 \times -6 \]
03

Simplify the multiplication

Multiply the numbers on each side of the equation: \[ 72q = -936 \]
04

Solve for q

Isolate q by dividing both sides of the equation by 72: \[ q = \frac{-936}{72} \]
05

Calculate the final value

Perform the division to find q: \[ q = -13 \]

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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 method used in algebra to solve proportions. A proportion is simply an equation stating that two ratios are equivalent. For example, given the proportion \(\frac{72}{156}=\frac{-6}{q}\), cross-multiplication can be used to find the value of the unknown variable q. This involves multiplying the numerator of the first ratio by the denominator of the second ratio and doing the same for the other pair of numerator and denominator. So, in this case, we multiply 72 by q and 156 by -6 to get the equation 72q = 156 脳 -6. This step transforms the proportion into a simple linear equation, making it easier to solve.
proportion equations
Proportion equations are equations that state two ratios are equal. They often appear as \(\frac{a}{b} = \frac{c}{d}\), where a, b, c, and d are numbers or expressions. Solving these equations usually involves finding the value of one of the variables. In our given problem, \(\frac{72}{156} = \frac{-6}{q}\), we are asked to solve for q. The essential steps to solve a proportion equation include setting up the equation, cross-multiplying to get rid of the fractions, simplifying if necessary, and then solving for the unknown variable. These steps help in getting a clear value for the unknown, ensuring both sides of the original proportion remain equivalent after solving.
division in algebra
Division in algebra is used to isolate the variables and solve equations. In the context of solving proportions, after cross-multiplying, you often end up with a simple linear equation that requires division. For example, in the initial problem \(\frac{72}{156} = \frac{-6}{q}\), after cross-multiplying to get 72q = -936, the next step is to isolate q by dividing both sides of the equation by 72. This process is straightforward: \[ q = \frac{-936}{72} \]. Performing this division gives us the value of q, which in this case is -13. This final division step ensures you've accurately solved for the variable, making sure the proportions are balanced.

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