If `5a+3b=35 and (a)/(b)=(2)/(3)`, what is the value of a?

Options

A. `(14)/(5)`
B. `(7)/(2)`
C. `5`
D. `7`

Answer

To solve the problem, we will follow these steps:

1. **Write down the equations**: We have two equations given:
\[ 5a + 3b = 35 \quad \text{(1)} \]
\[ \frac{a}{b} = \frac{2}{3} \quad \text{(2)} \]

2. **Express b in terms of a**: From equation (2), we can express \(b\) in terms of \(a\):
\[ a = \frac{2}{3}b \implies b = \frac{3}{2}a \quad \text{(3)} \]

3. **Substitute b into the first equation**: Now substitute equation (3) into equation (1):
\[ 5a + 3\left(\frac{3}{2}a\right) = 35 \]

4. **Simplify the equation**: Calculate \(3 \times \frac{3}{2}a\):
\[ 5a + \frac{9}{2}a = 35 \]

5. **Combine like terms**: To combine \(5a\) and \(\frac{9}{2}a\), convert \(5a\) to a fraction:
\[ 5a = \frac{10}{2}a \]
So,
\[ \frac{10}{2}a + \frac{9}{2}a = 35 \]
This simplifies to:
\[ \frac{19}{2}a = 35 \]

6. **Solve for a**: Multiply both sides by \(\frac{2}{19}\):
\[ a = 35 \times \frac{2}{19} \]
\[ a = \frac{70}{19} \]

7. **Final answer**: Therefore, the value of \(a\) is:
\[ a = \frac{70}{19} \]