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Chapter 8: Problem 148

Solve the equations for the variable. $$\frac{5}{4} c-3=\frac{1}{4} c-16$$

Short Answer

Expert verified
c = -13

Step by step solution

01

Isolate the variable term

First, we need to get all the terms involving the variable on one side of the equation. Subtract \( \frac{1}{4} c \) from both sides of the equation: \[ \frac{5}{4} c - \frac{1}{4} c - 3 = -16. \]
02

Simplify the equation

Combine the \( c \) terms on the left side: \[ \left( \frac{5}{4} - \frac{1}{4} \right) c - 3 = -16. \] This simplifies to \[ \frac{4}{4} c - 3 = -16, \] which further simplifies to \[ c - 3 = -16. \]
03

Solve for the variable

Now, isolate \( c \) by adding 3 to both sides: \[ c - 3 + 3 = -16 + 3, \] which simplifies to \[ c = -13. \]

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Key Concepts

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

Isolate the variable
When solving linear equations, the first step is to isolate the variable term on one side of the equation. This means moving all terms that contain the variable to one side and all constant terms (numbers without variables) to the other side.
In the given equation \[ \frac{5}{4} c - 3 = \frac{1}{4} c - 16 \], we start by subtracting \[ \frac{1}{4} c \] from both sides. This helps to gather all the \(c\) terms on one side, making the next steps easier. Thus, the equation transforms:
\[ \frac{5}{4} c - \frac{1}{4} c - 3 = -16 \].
By isolating the variable, we set ourselves up for easier manipulation in the following steps.
Combine like terms
Once the variable terms are on one side, the next step is to combine like terms. 'Like terms' are terms that have the same variable raised to the same power. In our equation, \[ \frac{5}{4} c - \frac{1}{4} c - 3 = -16 \], we recognize that \[ \frac{5}{4} c \] and \[ \frac{1}{4} c \] are like terms.
The fractions \[ \frac{5}{4} \] and \[ \frac{1}{4} \] can be combined by subtracting \[ \frac{1}{4} \] from \[ \frac{5}{4} \]:
\[ \left( \frac{5}{4} - \frac{1}{4} \right) c - 3 = -16 \]
This simplifies to:
\[ \frac{4}{4} c - 3 = -16 \],
which further simplifies to:
\[ c - 3 = -16 \].
Combining like terms reduces complexity and moves us closer to the solution.
Solve for the variable
The final step is to solve for the variable by isolating it completely. At this stage, we have a simplified equation with just one variable term. From the earlier steps, we have:
\[ c - 3 = -16 \]
Now, to isolate \( c \):
Add 3 to both sides of the equation:
\[ c - 3 + 3 = -16 + 3 \]
This simplifies to:
\[ c = -13 \]
Hence, after isolating the variable term, combining like terms, and solving for the variable, we determine that \( c \) equals -13. This methodical approach ensures accuracy and clarity in solving linear equations.

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Most popular questions from this chapter

In the following exercises, solve each equation. $$ 10 y=-5 y+60 $$

In the following exercises, solve each equation using the Multiplication Property of Equality. $$ 36=\frac{3}{4} x $$

In the following exercises, solve each equation by clearing the decimals. $$ 0.8 x-0.3=0.7 x+0.2 $$

In the following exercises, solve each linear equation using the general strategy. $$ \frac{1}{3}(6 m+21)=m-7 $$

In the following exercises, translate to an equation and then solve. The sum of \(y\) and \(-3\) is 40
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