Problem 29 - College Algebra

Question

Solve for the variable. $$ 4 y+8=2 y $$

Step-by-Step Solution

Answer

The solution is \( y = -4 \).

Step 1: Isolate terms with y on one side

First, we want to get all the terms involving the variable \( y \) on one side of the equation. To do this, subtract \( 2y \) from both sides:\[ 4y + 8 - 2y = 2y - 2y \] simplifies to:\[ 2y + 8 = 0 \]

Step 2: Move constant term to the other side

Next, we want to move the constant term (8) to the other side of the equation by subtracting 8 from both sides:\[ 2y + 8 - 8 = 0 - 8 \] which simplifies to:\[ 2y = -8 \]

Step 3: Solve for y

Now, divide both sides by 2 to solve for \( y \):\[ \frac{2y}{2} = \frac{-8}{2} \]This simplifies to:\[ y = -4 \]

Key Concepts

Isolating Variables

In solving linear equations, one of the fundamental skills you'll need is isolating the variable. This process involves manipulating the equation so that the variable you are solving for, in this case, \( y \), stands alone on one side of the equation.

For instance, in the problem \( 4y + 8 = 2y \), the variable \( y \) appears on both sides. To isolate \( y \), we start by eliminating one of those terms. Here, subtracting \( 2y \) from each side is a good first step.

Now you have:

• \( 4y - 2y + 8 = 0 \)
• This simplifies to \( 2y + 8 = 0 \)

This reduction process is essential because it simplifies the equation and helps highlight the isolated variable. Having \( y \) isolated makes it much easier to solve the equation. When isolating variables, aim for clarity and neatness; each step builds logically on the previous.

Balancing Equations

Balancing equations is a critical step in solving linear equations. The primary rule for balancing is that whatever operation you perform on one side, you must also perform on the other. This keeps the equation valid and ensures the equality is maintained.

Using the equation \( 2y + 8 = 0 \) obtained from the previous step, our goal was to isolate \( y \). This means we needed to balance the equation further by removing constants from the side containing \( y \).

• To do this, subtract \( 8 \) from both sides.
• We get: \( 2y = -8 \)

This subtraction step is a perfect example of balancing. The operations on both sides ensure that the equation remains true. Next, divide both sides by \( 2 \) to completely isolate \( y \):

• \( y = -4 \)

Balancing is essential because it is what keeps the integrity of the equation maintained while we work to isolate and solve for the variable.

Linear Equations

Linear equations are equations that graph as straight lines. They have variables raised only to the first power, and they can usually be written in the form \( ax + b = c \).

In our example, \( 4y + 8 = 2y \) is indeed a linear equation because \( y \) is the only variable and it's to the power of one.

• These equations are straightforward because they involve simple operations that change one side of the equation at a time.

Linear equations form the basis for more complex algebraic concepts. Understanding how to solve them, like we did by isolating \( y \), is crucial for progressing in algebra. By mastering linear equations, you'll gain a solid foundation for tackling more intricate mathematical problems involving different forms and complexities of equations.