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Chapter 4: Problem 55

Solve each equation for \(y\). $$ 9 x-2 y=4 $$

Short Answer

Expert verified
y = \frac{9x - 4}{2}

Step by step solution

01

Isolate the term with y

Begin by isolating the term with y on one side of the equation. Subtract 9x from both sides of the equation: \[9x-2y-9x=4-9x\]This simplifies to: \[-2y=4-9x\]
02

Solve for y

To solve for y, divide both sides of the equation by -2: \[y = \frac{4 - 9x}{-2}\]Simplify the right-hand side to get: \[y = -2 + \frac{9x}{2}\]
03

Simplify the equation

The final step is to further simplify the expression, if possible:The term \( - \frac{4}{-2} \) simplifies to 2, so: \[y = \frac{9x - 4}{2}\].

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Key Concepts

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

Isolate Variable
When solving linear equations, the first step is often to isolate the variable you are solving for. In this exercise, the goal is to isolate the term containing the variable \( y \).To start, we need to move all other terms to the opposite side of the equation.

In the equation \( 9x - 2y = 4 \), we need to isolate \( -2y \). This is done by subtracting \( 9x \) from both sides. The equation becomes: \[ -2y = 4 - 9x \]

This step makes the equation simpler and brings us closer to solving for \( y \). Isolating variables helps in focusing on the term you want to solve.
Solve for y
After isolating the term with \( y \), the next step is to solve for \( y \). The equation at this stage is: \[ -2y = 4 - 9x \]

To solve for \( y \), we divide both sides of the equation by the coefficient of \( y \), which in this case is \( -2 \). Doing this, we get: \[ y = \frac{4 - 9x}{-2} \]

Dividing both sides ensures that \( y \) is isolated by itself on one side of the equation. This step is crucial in linear equations as it directly gives the value of the variable we are interested in.
Simplify Expression
The final step in solving the equation is to simplify the expression to its neatest form. We have: \[ y = \frac{4 - 9x}{-2} \]

We can break this down further. Since we are dividing all parts of the numerator by \( -2 \), each term should be divided separately: \[ y = \frac{4}{-2} - \frac{9x}{-2} \]

Simplifying each term individually, we get: \[ y = -2 + \frac{9x}{2} \]

Making sure the equation is as simply written as possible is key to clarity and understanding. This final tidy-up ensures the solution is presented in the simplest form, making it easier to interpret.

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Most popular questions from this chapter

Solve each system by the elimination method. $$ \begin{aligned} &3 x-5 y=1\\\ &6 x-10 y=4 \end{aligned} $$

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