Certainly! Let's solve the equation step-by-step:
[tex]\[
\frac{8}{9}(54x - 36) + 2 = -\frac{3}{4}(-40 + 16x) + 90x
\][/tex]
First, we distribute the constants inside the parentheses:
On the left-hand side:
[tex]\[
\frac{8}{9}(54x - 36)
\][/tex]
Multiply inside the parentheses:
[tex]\[
\frac{8}{9} \cdot 54x - \frac{8}{9} \cdot 36 = 48x - 32
\][/tex]
So the left-hand side becomes:
[tex]\[
48x - 32 + 2 = 48x - 30
\][/tex]
Now, on the right-hand side:
[tex]\[
-\frac{3}{4}(-40 + 16x)
\][/tex]
Distribute the constant inside the parentheses:
[tex]\[
-\frac{3}{4} \cdot (-40) + \frac{3}{4} \cdot 16x = 30 + 12x
\][/tex]
So the right-hand side becomes:
[tex]\[
30 + 12x + 90x = 30 + 102x
\][/tex]
Now we have the equation:
[tex]\[
48x - 30 = 30 + 102x
\][/tex]
To solve for [tex]\(x\)[/tex], we first get all terms involving [tex]\(x\)[/tex] on one side and constants on the other side. Subtract [tex]\(48x\)[/tex] from both sides:
[tex]\[
-30 = 30 + 54x
\][/tex]
Then, subtract 30 from both sides:
[tex]\[
-60 = 54x
\][/tex]
Solve for [tex]\(x\)[/tex] by dividing both sides by 54:
[tex]\[
x = \frac{-60}{54} = -\frac{10}{9}
\][/tex]
So the solution is:
[tex]\[
x = -\frac{10}{9}
\][/tex]
