To solve the given equation for [tex]\( d \)[/tex], we can proceed as follows:
[tex]\[
\frac{3}{2}(7+3d) = 3 + 3d
\][/tex]
First, we distribute the [tex]\( \frac{3}{2} \)[/tex] on the left side of the equation:
[tex]\[
\frac{3}{2} \cdot (7 + 3d) = \frac{3}{2} \cdot 7 + \frac{3}{2} \cdot 3d = \frac{21}{2} + \frac{9d}{2}
\][/tex]
Thus, the equation now looks like:
[tex]\[
\frac{21}{2} + \frac{9d}{2} = 3 + 3d
\][/tex]
Next, to eliminate the fractions, we multiply every term in the equation by 2:
[tex]\[
2 \cdot \left( \frac{21}{2} + \frac{9d}{2} \right) = 2 \cdot 3 + 2 \cdot 3d
\][/tex]
This simplifies to:
[tex]\[
21 + 9d = 6 + 6d
\][/tex]
Now, we have a linear equation:
[tex]\[
21 + 9d = 6 + 6d
\][/tex]
To solve for [tex]\( d \)[/tex], we first get all the [tex]\( d \)[/tex]-terms on one side and the constants on the other side. Subtract [tex]\( 6d \)[/tex] from both sides:
[tex]\[
21 + 9d - 6d = 6 + 6d - 6d
\][/tex]
This simplifies to:
[tex]\[
21 + 3d = 6
\][/tex]
Next, we isolate the term with [tex]\( d \)[/tex] by subtracting 21 from both sides:
[tex]\[
3d = 6 - 21
\][/tex]
[tex]\[
3d = -15
\][/tex]
Finally, we solve for [tex]\( d \)[/tex] by dividing both sides by 3:
[tex]\[
d = \frac{-15}{3}
\][/tex]
[tex]\[
d = -5
\][/tex]
So, the value of [tex]\( d \)[/tex] is:
[tex]\[
d = -5
\][/tex]
