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Solve for \( x \):
[tex]\[ \frac{3(5x+4)}{9}+\frac{4(6x+4)}{8}=22 \][/tex]

Sagot :

GinnyAnswer
To solve the equation \(\frac{3(5x + 4)}{9} + \frac{4(6x + 4)}{8} = 22\), let's go through the steps in detail:

1. Clear the fractions:
To eliminate the denominators, we find the least common multiple (LCM) of 9 and 8. The LCM of 9 and 8 is 72. We will multiply every term by 72 to clear the fractions.

[tex]\[ 72 \cdot \left(\frac{3(5x + 4)}{9}\right) + 72 \cdot \left(\frac{4(6x + 4)}{8}\right) = 72 \cdot 22 \][/tex]

2. Simplify the equation:
We simplify each term by performing the multiplications:

[tex]\[ 72 \cdot \left(\frac{3(5x + 4)}{9}\right) = 8 \cdot 3(5x + 4) = 24(5x + 4) \][/tex]
[tex]\[ 72 \cdot \left(\frac{4(6x + 4)}{8}\right) = 9 \cdot 4(6x + 4) = 36(6x + 4) \][/tex]

Thus, the equation becomes:

[tex]\[ 24(5x + 4) + 36(6x + 4) = 72 \cdot 22 \][/tex]

3. Distribute within the parentheses:
Distribute the constants through the parentheses:

[tex]\[ 24(5x + 4) = 120x + 96 \][/tex]
[tex]\[ 36(6x + 4) = 216x + 144 \][/tex]

So, the equation is now:

[tex]\[ 120x + 96 + 216x + 144 = 1584 \][/tex]

4. Combine like terms:
Combine the \(x\) terms and the constant terms:

[tex]\[ 120x + 216x + 96 + 144 = 1584 \][/tex]
[tex]\[ 336x + 240 = 1584 \][/tex]

5. Isolate the \(x\) term:
Move the constant term to the other side of the equation:

[tex]\[ 336x = 1584 - 240 \][/tex]
[tex]\[ 336x = 1344 \][/tex]

6. Solve for \(x\):
Divide both sides of the equation by 336:

[tex]\[ x = \frac{1344}{336} \][/tex]

Thus,

[tex]\[ x = 4 \][/tex]

So, the solution to the equation [tex]\(\frac{3(5x + 4)}{9} + \frac{4(6x + 4)}{8} = 22\)[/tex] is [tex]\(x = 4\)[/tex].
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