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Chapter 6: Problem 56

Solve. $$\frac{12}{m}=\frac{18}{m+2}$$

Short Answer

Expert verified
The value of m that solves the equation is 4.

Step by step solution

01

Cross Multiply

To get rid of the fractions, cross-multiply. This means to multiply the denominator of the first fraction by the numerator of the second and vice versa. Hence we get \(12 * (m + 2) = 18 * m\). When we distribute the 12 and 18 to \(m\) and 2 respectively, we get \(12m + 24 = 18m\).
02

Isolate m

Next, to isolate m, subtract 12m from both sides of the equation. Hence, we have \(24 = 18m - 12m = 6m\).
03

Solve for m

Finally, to solve for the value of \(m\), divide both sides of the equation by 6. This gives us \(m = 24 / 6 = 4\).

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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 useful technique for solving equations that involve fractions. The main idea is to eliminate the fractions by multiplying across the equal sign.
This technique works when you have an equation of the form \(\frac{a}{b} = \frac{c}{d}\). The steps are simple: you multiply the numerator of the first fraction by the denominator of the second, and vice versa.
By cross-multiplying, you create a new equation without fractions: \(a \cdot d = b \cdot c\).
This technique is especially helpful because it turns a more complex equation into something easier to work with, often a simpler linear equation.
  • Easy and overcomes the need to find a common denominator.
  • Allows solving more quickly and with fewer steps.
  • Reduces the chance of making errors with fractions.
Solving for Variables
Once you have cross-multiplied and cleared the fractions, the next goal is to solve for the variable. Solving for a variable means you want this letter or symbol to be on one side of the equation alone.
In the equation \(12m + 24 = 18m\), we have the variable \(m\) on both sides. The first step is to get all terms involving \(m\) on one side.
Usually, you'll need to collect like terms, adding or subtracting as necessary.
  • This might involve moving terms from one side of the equation to the other.
  • In our example, we subtract \(12m\) from both sides, simplifying the equation.
  • This results in \(24 = 6m\), making the simplest form to work with.
This approach makes it easier to isolate and ultimately find the value of the variable.
Isolating Variables
Isolating the variable is the final step in solving the equation. This means getting your variable, such as \(m\), by itself on one side of the equation.
You want to obtain something like \(m =\) followed by its solution. In the equation where \(24 = 6m\), we need to isolate \(m\).
To do this, divide both sides of the equation by the coefficient of \(m\), which in this case is 6.
  • This step simplifies the equation, showing \(m = 4\).
  • Always perform the same operation on both sides to maintain balance.
  • Check your answer by substituting it back into the original equation.
Isolating the variable is crucial as it helps verify the solution and ensures that the solution is indeed correct.

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Most popular questions from this chapter

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are 0. SEE EXAMPLE 9. (OBJECTIVE 3) $$\frac{a b+b+2 a+2}{a b+a+b+1}$$

Explain the difference between a ratio and a proportion.

Set up and solve a proportion. A cologne can be made by mixing 3 drops of pure essence with 7 drops of distilled water. How many drops of water should be used with 42 drops of pure essence?

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{x y+3 y+3 x+9}{x^{2}-9}$$

Explain how to do the division \(\frac{a}{b} \div \frac{c}{d} \div \frac{e}{f}.\)
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