Certainly! Let's solve the linear equation [tex]\(3x + \frac{2}{3}y = 12\)[/tex] for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex].
1. Isolate the [tex]\(y\)[/tex]-term: To start, we need to first isolate the term involving [tex]\(y\)[/tex] on one side of the equation.
[tex]\[
3x + \frac{2}{3}y = 12
\][/tex]
2. Subtract [tex]\(3x\)[/tex] from both sides: By doing this, we can move the [tex]\(x\)[/tex]-term to the right side of the equation.
[tex]\[
\frac{2}{3}y = 12 - 3x
\][/tex]
3. Eliminate the fraction: To remove the fraction [tex]\(\frac{2}{3}\)[/tex], multiply every term by the reciprocal of [tex]\(\frac{2}{3}\)[/tex], which is [tex]\(\frac{3}{2}\)[/tex].
[tex]\[
y = (12 - 3x) \times \frac{3}{2}
\][/tex]
4. Distribute [tex]\(\frac{3}{2}\)[/tex] on the right-hand side: Multiply both terms inside the parentheses by [tex]\(\frac{3}{2}\)[/tex].
[tex]\[
y = 12 \times \frac{3}{2} - 3x \times \frac{3}{2}
\][/tex]
5. Simplify the multiplication:
[tex]\[
y = 18 - 4.5x
\][/tex]
Therefore, the solution for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex] is:
[tex]\[
y = 18 - 4.5x
\][/tex]
This equation tells us the value of [tex]\(y\)[/tex] for any given [tex]\(x\)[/tex] based on the provided linear relationship.
