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Chapter 2: Problem 38

Solve each formula for the indicated letter. $$ 5 x-4 y=8, \text { for } y $$

Short Answer

Expert verified
y = \frac{5}{4}x - 2

Step by step solution

01

- Isolate the term with y

Start with the equation: \[ 5x - 4y = 8 \] Subtract 5x from both sides to get the term with y on one side of the equation: \[ -4y = 8 - 5x \]
02

- Solve for y

Now, divide both sides by -4 to solve for y: \[ y = \frac{8 - 5x}{-4} \]
03

- Simplify the equation

Simplify the equation by dividing each term in the numerator by -4: \[ y = \frac{8}{-4} - \frac{5x}{-4} \] Which simplifies to: \[ y = -2 + \frac{5}{4}x \] or equivalently, \[ y = \frac{5}{4}x - 2 \]

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Key Concepts

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

Isolating Variables
One of the first steps in solving linear equations is to isolate the variable you are solving for. In the given problem, we want to solve for y.

We start with the equation: \ \[5x - 4y = 8\]
This equation has terms with x and y. To isolate y, we need to move all other terms to the other side by using basic arithmetic operations. By subtracting 5x from both sides: \ \[-4y = 8 - 5x\]
we get closer to isolating y. The subtraction helps us separate y from the x-term and constant on the other side of the equation.
Algebraic Manipulation
Algebraic manipulation involves performing mathematical operations to simplify and solve equations. Here, we will manipulate the equation further to solve for y.

In our example, we previously obtained: \ \[-4y = 8 - 5x\]
To solve for y, we need to remove the coefficient \-4\ from y by dividing both sides of the equation by \-4\. This yields: \ \[y = \frac{8 - 5x}{-4}\]
Next, we simplify by separating the numerator terms: \ \[y = \frac{8}{-4} - \frac{5x}{-4}\]
So, we end up with: \ \[y = -2 + \frac{5}{4}x\].
Thus, we have isolated y using algebraic manipulation techniques.
Step-by-Step Algebra
Using step-by-step algebra makes complex problems easier to solve. Let's recap the steps we took to solve this equation for y.

Step 1: Isolate the term with y by subtracting 5x from both sides: \ \[-4y = 8 - 5x\]
Step 2: Divide both sides by \-4\ to solve for y: \ \[y = \frac{8 - 5x}{-4}\]
Step 3: Simplify by dividing each term of the numerator by \-4\: \ \[y = -2 + \frac{5}{4}x\]
Step-by-step algebra ensures we handle each part systematically, making complex equations manageable and easier to understand.

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Most popular questions from this chapter

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