Problem 59 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ \frac{9}{2}+\frac{5}{2} y=2 y-4 $$

Step-by-Step Solution

Answer

The solution is \( y = -17 \).

Step 1: Eliminate Fractions

The equation is \( \frac{9}{2} + \frac{5}{2}y = 2y - 4 \). First, eliminate fractions by multiplying every term by 2 to get: \( 9 + 5y = 4y - 8 \).

Step 2: Rearrange Terms

Move all the terms involving \( y \) to one side and constants to the other side: \( 5y - 4y = -8 - 9 \), which simplifies to \( y = -17 \).

Step 3: Solution Check

Substitute \( y = -17 \) back into the original equation to verify the solution: \( \frac{9}{2} + \frac{5}{2}(-17) = 2(-17) - 4 \). This simplifies to \(-40 = -40 \), confirming the solution is correct.

Key Concepts

Solve Linear Equations

When solving linear equations, the primary goal is to find the value of the variable that makes the equation true. A linear equation is one where the variable, such as \( y \) in this instance, is raised to the power of one. The goal is to isolate this variable on one side of the equation.
To do this, you often need to rearrange terms on both sides of the equation by either adding, subtracting, multiplying, or dividing. In our example equation, we've managed to start with \( \frac{9}{2} + \frac{5}{2} y = 2y - 4 \).

• Identify like terms, which simply refers to terms that involve the variable \( y \) in our case, and constants.
• Rearrange the terms so that the \( y \) terms are on one side, while constants move to the other side.
• Simplify these terms to find the value of \( y \).

This process leads us to the solution, \( y = -17 \), which properly simplifies the equation.

Eliminate Fractions

Fractions can make linear equations more challenging to work with. Eliminating them helps simplify calculations. To remove fractions from a linear equation, find a common multiple of the denominators and multiply all terms by this number. This step ensures that fractions turn into whole numbers.
In the given equation, \( \frac{9}{2} + \frac{5}{2} y = 2y - 4 \), we have a denominator of 2. Multiplying every term by 2 eliminates the fractions and simplifies the equation to \( 9 + 5y = 4y - 8 \).

• Choose the least common multiple of the denominators.
• Multiply every term in the equation by that number.
• Transform the equation into a simpler form, without fractions.

By eliminating fractions, it's easier to manipulate and solve the equation as it reduces potential arithmetic errors.

Solution Verification

To ensure that a solution is correct, it is crucial to verify it by substituting the variable back into the original equation. This ensures no mistakes were made during the solving process.
For our equation, substituting \( y = -17 \) back gives us:

• On the left side: \( \frac{9}{2} + \frac{5}{2}(-17) = -40 \).
• On the right side: \( 2(-17) - 4 = -40 \).

Both sides equal \( -40 \), confirming that the solution is correct.
If both sides of the equation equate after substitution, you've found the correct solution. This verification step is vital as it confirms the accuracy of your computation and logical steps taken. It's always the closing part of solving any equation.