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Chapter 2: Problem 31

Solve for the specified variable. \(x y+5=y-7\) for \(y\)

Short Answer

Expert verified
y = \frac{-12}{x - 1}

Step by step solution

01

Isolate the variable terms

Begin by moving the terms involving the variable \(y\) to one side of the equation and all other terms to the opposite side. Subtract \(y\) from both sides.\(x y + 5 - y = y - 7 - y\)This simplifies to:\(x y - y + 5 = -7\)
02

Factor out the variable

Notice that both terms on the left side of the equation have a common variable factor \(y\). Factor out \(y\) from the left side.\(y(x - 1) + 5 = -7\)
03

Isolate the variable term

Subtract 5 from both sides to isolate the term with \(y\).\(y(x - 1) + 5 - 5 = -7 - 5\)This simplifies to:\(y(x - 1) = -12\)
04

Solve for the variable

Finally, divide both sides of the equation by \(x - 1\) to solve for \(y\).\(y = \frac{-12}{x - 1}\)

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Key Concepts

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

Isolating Variables
Isolating variables is a crucial step in solving equations. We want to get the variable alone on one side of the equation to find its value. In this exercise, our goal is to solve for the variable \( y \). Here are the steps we took:

First, we started with the equation: \( xy + 5 = y - 7 \).
To isolate \( y \), we need to move all terms involving \( y \) to one side and the remaining terms to the other side. This means we'll subtract \( y \) from both sides:

\( xy + 5 - y = y - 7 - y \)
This gives us:

\( xy - y + 5 = -7 \)
At this point, \( y \) is not isolated yet, but we have all terms with \( y \) on one side. Now we'll continue to the next step of factoring.
Factoring Expressions
Factoring expressions helps simplify equations when terms share a common factor. In our equation, \( xy - y + 5 = -7 \), both terms on the left side share the variable \( y \). We can factor \( y \) out to make the equation easier to solve:

\( y(x - 1) + 5 = -7 \)
We factored \( y \) from the terms \( xy \) and \( -y \). This results in a simpler expression inside the parentheses. Next, we need to isolate the term with \( y \) before we can solve for it.
Solving Linear Equations
Solving linear equations often involves isolating the variable and simplifying. From our factored expression \( y(x - 1) + 5 = -7 \), we subtract 5 from both sides to isolate the term with \( y \):

\( y(x - 1) + 5 - 5 = -7 - 5 \)
This simplifies to:

\( y(x - 1) = -12 \)
Finally, we solve for \( y \) by dividing both sides by \( x - 1 \):

\( y = \frac{-12}{x - 1} \)
Now, \( y \) is isolated and we have our solution. Remember, the goal is to isolate the variable step by step, using operations like addition, subtraction, multiplication, division, and factoring to simplify the equation.

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Most popular questions from this chapter

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