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Chapter 7: Problem 68

Solve. \(12+6 y=2 x,\) for \(y\)

Short Answer

Expert verified
\[y = \frac{x}{3} - 2\]

Step by step solution

01

- Isolate the term involving y

First, move the constant term to the other side of the equation. Subtract 12 from both sides: i.e. -> \[12 + 6y - 12 = 2x - 12\] => \[6y = 2x - 12\]
02

- Solve for y

Next, solve for y by dividing both sides of the equation by 6: -> \[y = \frac{2x - 12}{6}\]
03

- Simplify the Equation

Simplify the expression by separating the terms in the numerator: -> \[y = \frac{2x}{6} - \frac{12}{6}\] => \[y = \frac{x}{3} - 2\]

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Key Concepts

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

Isolating Variables
When solving linear equations, isolating the variable is a crucial step. This means you want to get the variable you are solving for by itself on one side of the equation.

In our example, we start with the equation: \(12 + 6y = 2x\)

To isolate the term involving \(y\), we need to move the constant on the same side as \(y\) to the other side of the equation. This is done by subtracting \(12\) from both sides:

\(12 + 6y - 12 = 2x - 12\)

This simplifies to:

\(6y = 2x - 12\)

Now, the term \(6y\) is isolated, which sets up the next step to solve for \(y\).
Simplification
Simplification makes equations easier to work with by reducing them to their simplest form.

After isolating \(6y\), the equation is:

\(6y = 2x - 12\)

To solve for \(y\), we need to simplify the equation further by dividing each term by \(6\):

\(y = \frac{2x - 12}{6}\)

By separating terms in the numerator, we achieve:

\(y = \frac{2x}{6} - \frac{12}{6}\)

This simplifies to:

\(y = \frac{x}{3} - 2\)

Thus, the simplified form of the equation expresses \(y\) in terms of \(x\).
Algebraic Manipulation
Algebraic manipulation involves rearranging equations using operations like addition, subtraction, multiplication, and division.

In our example, the algebraic manipulation started by isolating \(6y\) and then dividing through by \(6\):

\(6y = 2x - 12\)

Dividing the entire equation by \(6\) gives us:

\(y = \frac{2x - 12}{6}\)

Then, separating the terms in the fraction simplifies the equation further:

\(y = \frac{2x}{6} - \frac{12}{6}\)

Finally, simplifying the fractions provides the clean equation:

\(y = \frac{x}{3} - 2\)

Understanding these fundamental principles helps in handling more complex algebraic equations efficiently.

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