Answer :
To solve the equation [tex]\(\frac{45}{12} - \frac{1}{4} v = v + \frac{9}{4}\)[/tex], let's break it down step-by-step:
1. Simplify the Fraction: Simplify [tex]\(\frac{45}{12}\)[/tex] to a simpler fraction.
[tex]\[
\frac{45}{12} = \frac{15}{4}
\][/tex]
This simplifies since 45 divided by 3 is 15, and 12 divided by 3 is 4.
2. Rewrite the Equation: Substitute the simplified fraction back into the equation.
[tex]\[
\frac{15}{4} - \frac{1}{4} v = v + \frac{9}{4}
\][/tex]
3. Move Terms Involving [tex]\(v\)[/tex] Together: Start by adding [tex]\(\frac{1}{4} v\)[/tex] to both sides to get all terms involving [tex]\(v\)[/tex] on the right.
[tex]\[
\frac{15}{4} = v + \frac{1}{4} v + \frac{9}{4}
\][/tex]
4. Combine Like Terms: Combine [tex]\(v + \frac{1}{4}v\)[/tex] on the right side.
[tex]\[
\frac{15}{4} = \left(1 + \frac{1}{4}\right)v + \frac{9}{4}
\][/tex]
[tex]\[
\frac{15}{4} = \frac{5}{4}v + \frac{9}{4}
\][/tex]
5. Isolate the [tex]\(v\)[/tex] Term: Subtract [tex]\(\frac{9}{4}\)[/tex] from both sides to isolate the term with [tex]\(v\)[/tex].
[tex]\[
\frac{15}{4} - \frac{9}{4} = \frac{5}{4}v
\][/tex]
[tex]\[
\frac{6}{4} = \frac{5}{4}v
\][/tex]
6. Simplify the Left Side: Simplify [tex]\(\frac{6}{4}\)[/tex].
[tex]\[
\frac{6}{4} = \frac{3}{2}
\][/tex]
7. Solve for [tex]\(v\)[/tex]: Divide both sides by [tex]\(\frac{5}{4}\)[/tex].
[tex]\[
\frac{3}{2} \div \frac{5}{4} = v
\][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[
v = \frac{3}{2} \times \frac{4}{5} = \frac{12}{10}
\][/tex]
8. Simplify to Find the Final Answer: Simplify [tex]\(\frac{12}{10}\)[/tex].
[tex]\[
\frac{12}{10} = \frac{6}{5} = 1.2
\][/tex]
So, the value of [tex]\(v\)[/tex] is [tex]\(1.2\)[/tex].
1. Simplify the Fraction: Simplify [tex]\(\frac{45}{12}\)[/tex] to a simpler fraction.
[tex]\[
\frac{45}{12} = \frac{15}{4}
\][/tex]
This simplifies since 45 divided by 3 is 15, and 12 divided by 3 is 4.
2. Rewrite the Equation: Substitute the simplified fraction back into the equation.
[tex]\[
\frac{15}{4} - \frac{1}{4} v = v + \frac{9}{4}
\][/tex]
3. Move Terms Involving [tex]\(v\)[/tex] Together: Start by adding [tex]\(\frac{1}{4} v\)[/tex] to both sides to get all terms involving [tex]\(v\)[/tex] on the right.
[tex]\[
\frac{15}{4} = v + \frac{1}{4} v + \frac{9}{4}
\][/tex]
4. Combine Like Terms: Combine [tex]\(v + \frac{1}{4}v\)[/tex] on the right side.
[tex]\[
\frac{15}{4} = \left(1 + \frac{1}{4}\right)v + \frac{9}{4}
\][/tex]
[tex]\[
\frac{15}{4} = \frac{5}{4}v + \frac{9}{4}
\][/tex]
5. Isolate the [tex]\(v\)[/tex] Term: Subtract [tex]\(\frac{9}{4}\)[/tex] from both sides to isolate the term with [tex]\(v\)[/tex].
[tex]\[
\frac{15}{4} - \frac{9}{4} = \frac{5}{4}v
\][/tex]
[tex]\[
\frac{6}{4} = \frac{5}{4}v
\][/tex]
6. Simplify the Left Side: Simplify [tex]\(\frac{6}{4}\)[/tex].
[tex]\[
\frac{6}{4} = \frac{3}{2}
\][/tex]
7. Solve for [tex]\(v\)[/tex]: Divide both sides by [tex]\(\frac{5}{4}\)[/tex].
[tex]\[
\frac{3}{2} \div \frac{5}{4} = v
\][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[
v = \frac{3}{2} \times \frac{4}{5} = \frac{12}{10}
\][/tex]
8. Simplify to Find the Final Answer: Simplify [tex]\(\frac{12}{10}\)[/tex].
[tex]\[
\frac{12}{10} = \frac{6}{5} = 1.2
\][/tex]
So, the value of [tex]\(v\)[/tex] is [tex]\(1.2\)[/tex].
