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Chapter 2: Problem 89

Solve each equation, if possible. $$ \frac{1}{4}(10-2 y)=8 $$

Short Answer

Expert verified
The solution to the equation is \( y = -11 \).

Step by step solution

01

Eliminate the Fraction

To start solving the equation \( \frac{1}{4}(10 - 2y) = 8 \), multiply both sides by 4 to remove the fraction. This will give: \[ 10 - 2y = 32 \]
02

Isolate the Variable Term

Next, you need to isolate the term involving \( y \). Subtract 10 from both sides:\[ -2y = 22 \]
03

Solve for the Variable

Now, divide both sides by -2 to solve for \( y \):\[ y = -11 \]

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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. Such equations involve variables that you need to solve for, like finding the value of \( y \). These equations often appear in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. The goal is to determine the value of the variable that makes the equation true.

To solve, follow a series of steps:
  • Begin by simplifying both sides of the equation, if necessary.
  • Eliminate any fractions or parentheses to ease calculation.
  • Manipulate the equation to get all variable terms on one side and constant terms on the other.
  • Perform operations to isolate the variable and find its value.
These steps ensure a structured approach to finding solutions for different kinds of linear equations.
Eliminating Fractions
Fractions can make equations seem more complex than they are. So, a useful step in solving equations is eliminating these fractions. This makes it easy to handle the equation in simpler integer terms.

In our example, we started with \( \frac{1}{4}(10 - 2y) = 8 \). The fraction \( \frac{1}{4} \) makes the equation more complex to solve directly. The trick is to multiply both sides of the equation by the denominator. In this case, multiplying everything by 4 eliminates \( \frac{1}{4} \).
  • This helps in transforming the equation to \( 10 - 2y = 32 \).
  • It simplifies calculations and helps avoid arithmetic errors.
Once the fraction is eliminated, the equation becomes simpler, easing further solving steps.
Isolating Variables
Achieving a clear solution requires isolating the variable. This means arranging the equation to end up with the variable on one side and the numbers on the other.

Once the fraction is eliminated and we have \( 10 - 2y = 32 \), the next step is to move all constant terms to the opposite side of the equation. Subtract 10 from both sides, resulting in \( -2y = 22 \).
  • This step involves performing the same operation on both sides of the equation.
  • Keeping the equation balanced is crucial in algebra.
Now, finally divide by the coefficient of \( y \), which is -2, to get \( y = -11 \). This isolation step untangles the variable, providing the solution.

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