What value of x makes the equation true?
The correct answer and explanation is:
The given equation is: x+35=4x−73.\frac{x + 3}{5} = \frac{4x – 7}{3}.
Steps to Solve for xx:
- Eliminate the Fractions: Multiply through by the least common denominator (LCD) of the denominators 55 and 33, which is 1515: 15⋅x+35=15⋅4x−73.15 \cdot \frac{x + 3}{5} = 15 \cdot \frac{4x – 7}{3}. Simplify each term: 3(x+3)=5(4x−7).3(x + 3) = 5(4x – 7).
- Distribute: Expand both sides: 3x+9=20x−35.3x + 9 = 20x – 35.
- Isolate xx: Bring all terms involving xx to one side and constants to the other: 3x−20x=−35−9.3x – 20x = -35 – 9. Simplify: −17x=−44.-17x = -44.
- Solve for xx: Divide both sides by −17-17: x=−44−17.x = \frac{-44}{-17}. Simplify: x=4417.x = \frac{44}{17}.
Final Answer:
x=4417.x = \frac{44}{17}.
Explanation :
To solve the equation x+35=4x−73\frac{x + 3}{5} = \frac{4x – 7}{3}, the goal is to find the value of xx that satisfies this equality. Since the equation involves fractions, the first step is to eliminate the denominators by multiplying through by their least common denominator (LCD). The LCD of 55 and 33 is 1515. Multiplying each term by 1515 ensures the fractions are cleared, simplifying the equation to 3(x+3)=5(4x−7)3(x + 3) = 5(4x – 7).
Next, the distributive property is applied to expand both sides. This results in 3x+9=20x−353x + 9 = 20x – 35. The equation now looks simpler, involving just linear terms and constants. To isolate xx, move all terms containing xx to one side of the equation and constants to the other. This rearrangement gives −17x=−44-17x = -44.
Dividing both sides by −17-17 provides the solution x=4417x = \frac{44}{17}. The value 4417\frac{44}{17} is a fraction that cannot be simplified further, so it is the final answer.
This method ensures the solution is exact. Verifying the solution involves substituting x=4417x = \frac{44}{17} back into the original equation. Both sides will equal when simplified, confirming the correctness of the solution. Understanding these steps is crucial for solving similar equations systematically.