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Chapter 2: Problem 20

For each formula, express y as a function of x. $$4 y+5 x=8$$

Short Answer

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

Step by step solution

01

Isolate the term with y

Start by moving all the terms involving y to one side of the equation and the rest to the other. The original equation is: \[ 4y + 5x = 8 \] Subtract \(5x\) from both sides: \[ 4y = 8 - 5x \]
02

Solve for y

To solve for y, divide both sides of the equation by 4: \[ y = \frac{8 - 5x}{4} \]
03

Simplify the expression

Simplify the right-hand side of the equation: \[ y = 2 - \frac{5}{4} x \]

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Key Concepts

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

solving for y
When working with an equation like \(4y + 5x = 8\), one of the first goals is to solve for \(y\). This means we need to isolate \(y\) on one side of the equation, while everything else stays on the other side.
This process often involves performing a series of algebraic manipulations to get \(y\) alone. Think of it as peeling away the layers of an onion until only \(y\) is left.
By solving for \(y\), we transform the equation into the form \(y = f(x)\), where \(f(x)\) is some function involving \(x\).
isolating variables
Isolating variables is a crucial step in solving equations. In our example, we start with \(4y + 5x = 8\). To isolate \(y\), we need to get rid of the \(5x\) term that is attached to it.
We do this by subtracting \(5x\) from both sides of the equation: \[ 4y = 8 - 5x \] Now, we've moved the \(5x\) to the right side and \(4y\) is isolated on the left side.
Next, we'll need to get \(y\) by itself, which requires another step of algebraic manipulation.
algebraic manipulation
Algebraic manipulation involves using mathematical operations to rearrange an equation. Once we've isolated \(4y\), we need to solve for \(y\) by dividing both sides by 4: \[ y = \frac{8 - 5x}{4} \]
This step-by-step manipulation follows the basic principles of algebra, ensuring we do the same operation to both sides of the equation.
By dividing both sides by 4, we balance the equation, keeping it true while isolating \(y\).
functions
Expressing \(y\) as a function of \(x\) means writing \(y\) in terms of \(x\). In this case, the function is \( y = 2 - \frac{5}{4}x \).
Function notation helps us see how \(y\) changes in response to different values of \(x\). Here, for any value of \(x\), we can plug it into the function and find the corresponding \(y\).
This functional relationship is key in mathematics because it shows direct dependence between variables and can be used in graphing and further analysis.

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Most popular questions from this chapter

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