Question
(23) \( \frac{1}{3}(2 x-1)=\frac{3}{4}(x+2) \)
Ask by Mckenzie Morrison. in the United States
Jan 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -22 \)
Solution
To solve the equation:
\[
\frac{1}{3}(2x - 1) = \frac{3}{4}(x + 2)
\]
**Step 1: Eliminate the Fractions**
Find the least common multiple (LCM) of the denominators 3 and 4, which is 12. Multiply both sides of the equation by 12 to eliminate the fractions:
\[
12 \cdot \frac{1}{3}(2x - 1) = 12 \cdot \frac{3}{4}(x + 2)
\]
Simplify each side:
\[
4(2x - 1) = 9(x + 2)
\]
**Step 2: Distribute the Multiplications**
Expand both sides of the equation:
\[
8x - 4 = 9x + 18
\]
**Step 3: Solve for \( x \)**
First, subtract \( 8x \) from both sides to get all \( x \)-terms on one side:
\[
-4 = x + 18
\]
Next, subtract 18 from both sides to isolate \( x \):
\[
-4 - 18 = x
\]
\[
-22 = x
\]
**Solution:**
\[
x = -22
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( \frac{1}{3}(2x-1)=\frac{3}{4}(x+2) \), first eliminate the fractions by finding a common multiple of 3 and 4, which is 12. Multiply both sides by 12 to clear the fractions: \[ 12 \cdot \frac{1}{3}(2x-1) = 12 \cdot \frac{3}{4}(x+2) \] This simplifies to: \[ 4(2x-1) = 9(x+2) \] Next, distribute both sides: \[ 8x - 4 = 9x + 18 \] Now, rearrange the equation by moving all terms involving \( x \) to one side and constant terms to the other: \[ 8x - 9x = 18 + 4 \] This gives: \[ -x = 22 \] Finally, multiply both sides by -1 to solve for \( x \): \[ x = -22 \] So the solution to the equation is \( x = -22 \).
