Problem 15 - College Algebra with Corequisite Support

Question

For each of the following exercises, solve the equation for y in terms of \(x\). $$ 5 y+4=10 x $$

Step-by-Step Solution

Answer

\(y = 2x - \frac{4}{5}\)

Step 1: Isolate the y-term

Start by isolating the term that contains \(y\). Given the equation \(5y + 4 = 10x\), subtract \(4\) from both sides to get \(5y = 10x - 4\).

Step 2: Solve for y

To solve for \(y\), divide each term in the equation \(5y = 10x - 4\) by \(5\): \(y = \frac{10x}{5} - \frac{4}{5}\).

Step 3: Simplify the equation

Simplify each fraction: \(y = 2x - \frac{4}{5}\).Thus, the solution is \(y = 2x - \frac{4}{5}\).

Key Concepts

Solving for a Variable

Solving for a variable means finding the value or expression for one variable in an equation. Here, we want the expression for \(y\) in terms of \(x\).
When you are tasked with solving for a variable, your goal is to rearrange the equation until the variable you're solving for is isolated on one side of the equation.
Let's break down the steps using our example, \(5y + 4 = 10x\):

• We want \(y\) by itself. That's the main idea.
• First, we eliminate any numbers or terms added or subtracted to \(y\). We do this by doing the opposite operation, using addition or subtraction.
• Next, if \(y\) is multiplied by a coefficient, divide everything by that coefficient. This gets \(y\) alone on one side.

By following these steps, we transformed the equation into \(y = 2x - \frac{4}{5}\), giving us \(y\) explicitly in terms of \(x\).
The process of solving for a variable is a foundational skill in algebra that you'll use again and again.

Linear Equations

Linear equations are equations of the first degree, which means they have no exponents higher than one. These equations graph as straight lines on a coordinate plane.
In our exercise, the equation \(5y + 4 = 10x\) is a classic linear equation. When put into a format like \(y = 2x - \frac{4}{5}\), it shows a linear relationship between \(x\) and \(y\).
Linear equations typically look like \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants:

• They represent relationships with constant rates of change.
• If you graph them, you will always get a smooth, straight line.
• Such equations help us understand direct relationships between different quantities.

Understanding linear equations is beneficial because they are everywhere in real-life problems, from calculating distances to managing finances.
Recognizing them and knowing how to solve them can make these practical situations much easier to handle.

Isolation of Variables

Isolation of variables is all about getting one variable alone on one side of the equation. This step is crucial when solving algebraic equations. For the equation \(5y + 4 = 10x\), our goal was to isolate \(y\).
Here's how you can think of isolation:

• Start with balance. You can do anything to one side as long as you do the exact same to the other side of the equation.
• Use inverse operations to get rid of numbers or terms around the variable. For example, we subtracted \(4\) to "move" it away from \(y\).
• Divide or multiply to remove any coefficients linked to the variable.

Isolation helps use the flexibility of algebra to rearrange equations according to needs, making it easier to understand relationships between variables.
It's an essential skill, whether you're dealing with simple tasks or complex mathematical problems.