Question $\frac{3}{4}(2 x-3)=\frac{6}{5}(3 x+1)$

Studdy Solution

STEP 1

1. The given problem is an equation that involves fractions and linear terms in $x$ .
2. The goal is to solve for $x$ .
3. Both sides of the equation can be simplified and cleared of fractions to isolate $x$ .

STEP 2

1. Distribute the constants on both sides of the equation.
2. Clear the fractions by finding a common multiple and multiplying both sides of the equation by that multiple.
3. Combine like terms to isolate $x$ .
4. Solve for $x$ .

STEP 3

Distribute the constants $\frac{3}{4}$ and $\frac{6}{5}$ on both sides of the equation.
$\frac{3}{4}(2x - 3) = \frac{6}{5}(3x + 1)$
Distribute $\frac{3}{4}$ on the left side: $\frac{3}{4} \cdot 2x - \frac{3}{4} \cdot 3 = \frac{3}{2}x - \frac{9}{4}$
Distribute $\frac{6}{5}$ on the right side: $\frac{6}{5} \cdot 3x + \frac{6}{5} \cdot 1 = \frac{18}{5}x + \frac{6}{5}$
The equation now looks like: $\frac{3}{2}x - \frac{9}{4} = \frac{18}{5}x + \frac{6}{5}$

STEP 4

Clear the fractions by finding a common multiple. The least common multiple (LCM) of 2, 4, and 5 is 20. Multiply both sides of the equation by 20.
$20 \left( \frac{3}{2}x - \frac{9}{4} \right) = 20 \left( \frac{18}{5}x + \frac{6}{5} \right)$

STEP 5

Multiply each term inside the parentheses by 20.
$20 \cdot \frac{3}{2}x - 20 \cdot \frac{9}{4} = 20 \cdot \frac{18}{5}x + 20 \cdot \frac{6}{5}$
Simplify each term: $10 \cdot 3x - 5 \cdot 9 = 4 \cdot 18x + 4 \cdot 6$
This simplifies to: $30x - 45 = 72x + 24$

STEP 6

Combine like terms to isolate $x$ . First, subtract $30x$ from both sides.
$30x - 45 - 30x = 72x + 24 - 30x$
This simplifies to: $-45 = 42x + 24$

STEP 7

Subtract 24 from both sides to further isolate $x$ .
$-45 - 24 = 42x$
This simplifies to: $-69 = 42x$

STEP 8

Divide both sides by 42 to solve for $x$ .
$x = \frac{-69}{42}$
Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3.
$x = \frac{-69 \div 3}{42 \div 3} = \frac{-23}{14}$
Solution: $x = \frac{-23}{14}$

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