\frac{7x+14}{3} – \frac{17-3x}{5} = \frac{6x-4x+2-5}{3}
The Correct Answer and Explanation is:
To solve the given equation: 7x+143−17−3×5=6x−4x+2−53\frac{7x + 14}{3} – \frac{17 – 3x}{5} = \frac{6x – 4x + 2 – 5}{3}37x+14−517−3x=36x−4x+2−5
Step 1: Simplify the right-hand side.
Start by simplifying the expression on the right side. Combine like terms: 6x−4x+2−5=2x−36x – 4x + 2 – 5 = 2x – 36x−4x+2−5=2x−3
So the equation becomes: 7x+143−17−3×5=2x−33\frac{7x + 14}{3} – \frac{17 – 3x}{5} = \frac{2x – 3}{3}37x+14−517−3x=32x−3
Step 2: Eliminate the denominators.
To eliminate the fractions, multiply through by the least common denominator (LCD) of 3 and 5, which is 15. Multiply both sides of the equation by 15: 15(7x+143)−15(17−3×5)=15(2x−33)15 \left( \frac{7x + 14}{3} \right) – 15 \left( \frac{17 – 3x}{5} \right) = 15 \left( \frac{2x – 3}{3} \right)15(37x+14)−15(517−3x)=15(32x−3)
Step 3: Simplify the equation.
Now simplify each term:
- For the first term: 15×7x+143=5(7x+14)=35x+7015 \times \frac{7x + 14}{3} = 5(7x + 14) = 35x + 7015×37x+14=5(7x+14)=35x+70
- For the second term: 15×17−3×5=3(17−3x)=51−9×15 \times \frac{17 – 3x}{5} = 3(17 – 3x) = 51 – 9×15×517−3x=3(17−3x)=51−9x
- For the third term: 15×2x−33=5(2x−3)=10x−1515 \times \frac{2x – 3}{3} = 5(2x – 3) = 10x – 1515×32x−3=5(2x−3)=10x−15
So the equation now becomes: 35x+70−(51−9x)=10x−1535x + 70 – (51 – 9x) = 10x – 1535x+70−(51−9x)=10x−15
Step 4: Distribute the negative sign.
Distribute the negative sign across the parentheses: 35x+70−51+9x=10x−1535x + 70 – 51 + 9x = 10x – 1535x+70−51+9x=10x−15
Simplify the left side: 35x+9x+70−51=10x−1535x + 9x + 70 – 51 = 10x – 1535x+9x+70−51=10x−15 44x+19=10x−1544x + 19 = 10x – 1544x+19=10x−15
Step 5: Solve for xxx.
Move all terms involving xxx to one side and constants to the other side: 44x−10x=−15−1944x – 10x = -15 – 1944x−10x=−15−19
Simplify: 34x=−3434x = -3434x=−34
Now divide both sides by 34: x=−3434x = \frac{-34}{34}x=34−34 x=−1x = -1x=−1
Final Answer:
The solution to the equation is x=−1x = -1x=−1
