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Chapter 8: Problem 305

In the following exercises, solve. $$ \frac{5}{2}-\frac{1}{c}=\frac{3}{4} $$

Short Answer

Expert verified
The value of \( c \) is \( \frac{4}{7} \).

Step by step solution

01

Isolate the Term with the Variable

First, subtract \( \frac{5}{2} \) from both sides of the equation to isolate the term with the variable \( c \).\[ \frac{5}{2} - \frac{1}{c} - \frac{5}{2} = \frac{3}{4} - \frac{5}{2} \] This simplifies to: \[ - \frac{1}{c} = \frac{3}{4} - \frac{10}{4} \]
02

Simplify the Right Side

Combine the fractions on the right-hand side: \[ \frac{3}{4} - \frac{10}{4} = \frac{3 - 10}{4} = \frac{-7}{4} \] This gives us: \[ - \frac{1}{c} = \frac{-7}{4} \]
03

Solve for the Variable

Multiply both sides by \( -1 \) to get rid of the negative sign: \[ \frac{1}{c} = \frac{7}{4} \] Now take the reciprocal of both sides to solve for \( c \): \[ c = \frac{4}{7} \]

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Key Concepts

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

Isolate the Variable
Isolating the variable means getting the variable by itself on one side of the equation. This helps simplify the solving process. For example, in the equation \(\frac{5}{2} - \frac{1}{c} = \frac{3}{4}\), the variable is \(c\). Our goal is to manipulate the equation so that \(c\) is alone on one side.
We do this step-by-step:
  • Start by subtracting \( \frac{5}{2} \) from both sides of the equation:

    • \[ \frac{5}{2} - \frac{1}{c} - \frac{5}{2} = \frac{3}{4} - \frac{5}{2} \]
    This helps move the fraction involving \(c\) to one side by itself. After subtracting \( \frac{5}{2} \) from both sides, the equation simplifies to:
    \[ - \frac{1}{c} = \frac{3}{4} - \frac{10}{4} \]
    This is the first step in solving the equation where the variable is isolated on one side!
Combine Fractions
Combining fractions means bringing them to a common denominator and then combining the numerators. This simplifies the expression and makes it easier to handle. For example, after isolating the variable term, you might end up with something like:
\[ - \frac{1}{c} = \frac{3}{4} - \frac{10}{4} \]
Here, both fractions already have the same denominator (4). Therefore, you can combine them easily by subtracting the numerators:
\[ \frac{3 - 10}{4} = \frac{-7}{4} \]
This simplifies the equation to:
\[ - \frac{1}{c} = \frac{-7}{4} \]
Combining fractions makes complex equations look simpler and easier to handle. This step is crucial to proceed with any further calculations effectively.
Reciprocal
To solve for the variable in some rational equations, you may need to use the concept of the reciprocal. The reciprocal of a number is what you multiply by to get 1. For example, the reciprocal of \( \frac{1}{c} \) is \( c \). In our example:
  • After simplifying the equation, we have:
    \[ - \frac{1}{c} = \frac{-7}{4} \]
  • We multiply both sides by -1:

    • \[ \frac{1}{c} = \frac{7}{4} \]
    To solve for \( c \), you take the reciprocal of both sides:
    \[ c = \frac{4}{7} \]
    This step helps find the final value of the variable. By understanding how to use the reciprocal, you can transform and solve equations easily. This is a common method in algebra, especially when dealing with fractions.

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Most popular questions from this chapter

In the following exercises, solve. The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for \(\$ 39.80\) (a) Write the equation that relates the price to the number of gallons. (b) How much will it cost Jesse for 15 gallons of gas?

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In the following exercises, solve. The distance an object falls varies directly to the square of the time it falls. A ball falls 72 feet in 3 seconds, (a) Write the equation that relates the distance to the time. (b) How far will the ball have fallen in 8 seconds?

If \(v\) varies inversely with \(w\) and \(v=18\) when \(w=\frac{1}{3}\) find the equation that relates \(v\) and \(w\).

If \(y\) varies inversely with \(x\) and \(y=11\) when \(x=3\) find the equation that relates \(x\) and \(y\).
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