Question

linear equations
solving a proportion of the form (x+a)/b = c/d
solve for v.
\\(\frac{5}{8} = \frac{v + 3}{12}\\)
simplify your answer as much as possible.
\\(v = \square\\)

Explanation:

Step1: Cross - multiply the proportion

To solve the proportion \(\frac{5}{8}=\frac{v + 3}{12}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \(5\times12 = 8\times(v + 3)\).
Calculating the left - hand side: \(5\times12=60\), and the right - hand side is \(8(v + 3)=8v+24\). So, the equation becomes \(60 = 8v + 24\).

Step2: Isolate the variable term

Subtract 24 from both sides of the equation \(60=8v + 24\) to isolate the term with \(v\).
\(60-24=8v+24 - 24\).
Calculating the left - hand side: \(60 - 24 = 36\), and the right - hand side simplifies to \(8v\). So, we have \(36 = 8v\).

Step3: Solve for v

Divide both sides of the equation \(36 = 8v\) by 8 to solve for \(v\).
\(v=\frac{36}{8}\).
Simplify the fraction \(\frac{36}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. \(\frac{36\div4}{8\div4}=\frac{9}{2}=4.5\)

Answer:

\(v=\frac{9}{2}\) (or \(v = 4.5\))