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Chapter 2: Problem 88

Solve for the indicated variable. $$ \text { Solve for } y: 34 x-15 y=12 $$

Short Answer

Expert verified
The solution is \( y = \frac{34}{15}x - \frac{4}{5} \).

Step by step solution

01

Isolate the Term with Variable y

We start with the equation: \[34x - 15y = 12.\]Our goal is to isolate the term containing the variable \(y\). To do this, subtract \(34x\) from both sides of the equation:\[-15y = 12 - 34x.\]
02

Solve for y

With \(-15y\) isolated, divide every term by \(-15\) to solve for \(y\):\[y = \frac{12 - 34x}{-15}.\]
03

Simplify the Expression

Simplify the expression for \(y\) by dividing each term in the numerator by \(-15\):\[y = \frac{12}{-15} + \frac{-34x}{-15}.\]This simplifies further to:\[y = -\frac{4}{5} + \frac{34}{15}x.\]

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Key Concepts

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

Isolating Variables
In the world of algebra, isolating a variable is a fundamental technique used to solve equations. Imagine you're given an equation like \(34x - 15y = 12\); your task is to solve for \(y\). To do this, you first need to isolate the term containing \(y\) on one side of the equation.
Start by moving all other terms to the opposite side. For instance, subtract the entire \(34x\) from both sides:
  • Original: \( 34x - 15y = 12 \)
  • Action: Subtract \(34x\)
  • Result: \(-15y = 12 - 34x\)
This step clears the path to focus solely on the variable \(y\), getting it closer to where you can "solve for \(y\)" in the next steps.
Remember, whatever operation you perform should keep the equation balanced鈥攚hat you do to one side, you must do to the other.
Solving for a Variable
Once you've isolated the term with the variable you're solving for, the next step is actually solving the equation for that variable. Continuing with our example, once \(-15y\) stands alone to one side of the equation:\(-15y = 12 - 34x\),we now want to make \(y\) the subject of the formula.

To solve for \(y\), divide all terms in the equation by \(-15\), the coefficient of \(y\). This gives:
  • \(y = \frac{12 - 34x}{-15}\)
By doing this, you effectively "undo" the multiplication that \(y\) was involved in, leaving \(y\) by itself on one side of the equation. This is an important step because it transforms the expression into a form that provides a direct solution for the variable \(y\).
Simplifying Expressions
After solving for a variable, you often have an expression that can be simplified. Simplification makes the expression easier to understand and work with. You've reached the equation:\(y = \frac{12 - 34x}{-15}\).While this is correct, it can be simplified for clarity.

Break up the fraction by dividing both parts of the numerator separately by \(-15\):
  • \(y = \frac{12}{-15} + \frac{-34x}{-15}\)
Now simplify the fractions individually:
  • \(\frac{12}{-15} = -\frac{4}{5}\), since both \(12\) and \(15\) are divisible by \(3\).
  • \(\frac{-34}{-15} = \frac{34}{15}\), since dividing by a negative changes the sign twice, canceling it out into a positive.
Combine these results for the simplified expression:\(y = -\frac{4}{5} + \frac{34x}{15}\).The simplified expression not only looks cleaner but is also ready for further algebraic manipulation, if needed.

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