Question

Solve the following equation using the distributive property: 17/3 (x-3/2) =-5/4

Answer 1

To solve the equation \( \frac{17}{3} \left( x - \frac{3}{2} \right) = -\frac{5}{4} \) using the distributive property, we first distribute \( \frac{17}{3} \) to both terms inside the parentheses.

\[
\frac{17}{3} x - \frac{17}{3} \cdot \frac{3}{2} = -\frac{5}{4}
\]

Calculating \( \frac{17}{3} \cdot \frac{3}{2} \):

\[
\frac{17 \cdot 3}{3 \cdot 2} = \frac{17}{2}
\]

Substituting this back into the equation, we have:

\[
\frac{17}{3} x - \frac{17}{2} = -\frac{5}{4}
\]

Next, we want to isolate \( \frac{17}{3} x \). To do this, we can add \( \frac{17}{2} \) to both sides:

\[
\frac{17}{3} x = -\frac{5}{4} + \frac{17}{2}
\]

To add these fractions, we need a common denominator. The least common multiple of \( 4 \) and \( 2 \) is \( 4 \). We can convert \( \frac{17}{2} \) to have this common denominator:

\[
\frac{17}{2} = \frac{17 \cdot 2}{2 \cdot 2} = \frac{34}{4}
\]

Now, substituting this back in, we have:

\[
\frac{17}{3} x = -\frac{5}{4} + \frac{34}{4}
\]

Now we can combine the fractions on the right:

\[
\frac{17}{3} x = \frac{34 - 5}{4} = \frac{29}{4}
\]

Next, we can solve for \( x \) by multiplying both sides of the equation by the reciprocal of \( \frac{17}{3} \), which is \( \frac{3}{17} \):

\[
x = \frac{29}{4} \cdot \frac{3}{17}
\]

Calculating this gives:

\[
x = \frac{29 \cdot 3}{4 \cdot 17} = \frac{87}{68}
\]

Thus, the solution to the equation \( \frac{17}{3} \left( x - \frac{3}{2} \right) = -\frac{5}{4} \) is:

\[
\boxed{\frac{87}{68}}
\]