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Chapter 2: Problem 52

Solve each equation, if possible. $$ \frac{x}{5}-5=-12 $$

Short Answer

Expert verified
The solution is \( x = -35 \).

Step by step solution

01

Isolate the variable term

To start solving the equation \( \frac{x}{5} - 5 = -12 \), we need to isolate the fraction that contains the variable. First, add 5 to both sides of the equation to remove \( -5 \) from the left side:\[ \frac{x}{5} - 5 + 5 = -12 + 5 \] This simplifies to:\[ \frac{x}{5} = -7 \]
02

Solve for x by eliminating the fraction

Now that you have \( \frac{x}{5} = -7 \), eliminate the fraction by multiplying both sides of the equation by 5. This will leave \( x \) by itself on the left side:\[ 5 \times \frac{x}{5} = -7 \times 5 \] This simplifies to:\[ x = -35 \]

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Key Concepts

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

Solving Linear Equations
Solving linear equations is a fundamental skill in algebra that involves finding the value of the unknown variable which makes the equation true. A "linear equation" generally takes the form \( ax + b = c \) where \( x \) is the variable, and \( a \), \( b \), and \( c \) are constants. In essence, your goal is to "balance" the equation by performing the same operation on both sides until you solve for \( x \).Approach linear equations in these steps:
  • Simplify: Combine like terms on both sides if needed.
  • Isolate the variable: Use inverse operations to get \( x \) alone.
  • Check your solution: Substitute back to confirm correctness.
Remember that what you do to one side of the equation, you must do to the other. This maintains the balance of the equation.
Isolate the Variable
Isolating the variable is key when solving equations. This means getting \( x \) on its own on one side of the equation. Let's take our equation:\[ \frac{x}{5} - 5 = -12 \]Start by removing any constants from the same side as the variable fraction. Here, add 5 to both sides:\[ \frac{x}{5} = -7 \]Let's break down crucial points about isolation:
  • Focus on Addition/Subtraction: Get rid of numbers directly added or subtracted from \( x \) first.
  • Use Inverse Operations: If the equation has \( -5 \), add 5 to both sides.
Mastering the art of isolating the variable involves recognizing the sequence of operations applied to \( x \) and reversing them.
Eliminating Fractions
Fractions can make working with equations daunting, but eliminating them simplifies matters considerably. For the equation \( \frac{x}{5} = -7 \), eliminating the fraction means dealing with \( \frac{1}{5} \) on \( x \), effectively multiplying each side by 5.Let's execute that:\[ 5 \times \frac{x}{5} = -7 \times 5 \]This reduces down to \( x = -35 \). Key steps include:
  • Multiply by the Denominator: Apply this to both sides to "clear" the fraction.
  • Cross Check: Ensure all terms are accounted for when multiplying.
Mastering fractions gives you the confidence to tackle more complex equations where fractions frequently appear.

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