Let's solve the equation [tex]\( x + 9 = \frac{7}{4}x \)[/tex] for [tex]\( x \)[/tex].
1. Start with the given equation:
[tex]\[
x + 9 = \frac{7}{4}x
\][/tex]
2. To solve for [tex]\( x \)[/tex], we need to get all [tex]\( x \)[/tex]-terms on one side of the equation. Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
x + 9 - x = \frac{7}{4}x - x
\][/tex]
Simplifying the left side, we get:
[tex]\[
9 = \frac{7}{4}x - x
\][/tex]
3. Combine the [tex]\( x \)[/tex]-terms on the right side. Recall that [tex]\( x \)[/tex] is equivalent to [tex]\( \frac{4}{4}x \)[/tex]:
[tex]\[
9 = \frac{7}{4}x - \frac{4}{4}x
\][/tex]
Simplify by combining the fractions:
[tex]\[
9 = \left(\frac{7}{4} - \frac{4}{4}\right)x
\][/tex]
[tex]\[
9 = \frac{3}{4}x
\][/tex]
4. Now we need to solve for [tex]\( x \)[/tex]. To isolate [tex]\( x \)[/tex], multiply both sides of the equation by the reciprocal of [tex]\( \frac{3}{4} \)[/tex], which is [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[
x = 9 \cdot \frac{4}{3}
\][/tex]
5. Perform the multiplication:
[tex]\[
x = 9 \cdot \frac{4}{3} = 9 \cdot \frac{4}{3} = \frac{9 \cdot 4}{3} = \frac{36}{3} = 12
\][/tex]
So, the solution to the equation [tex]\( x + 9 = \frac{7}{4}x \)[/tex] is:
[tex]\[
x = 12
\][/tex]
