# Problem **Solve step-by-step:** $$ \frac{2x+3}{5} - \frac{3x-7}{2} = \frac{7x+1}{4} $$ --- # Step 1: Find the Least Common Denominator (LCD) The denominators are: 5,2, and 4. - LCD of 5, 2, and 4 is **20**. --- # Step 2: Multiply Both Sides by LCD Multiply both sides by 20 to clear denominators: \[ 20 \left[\frac{2x+3}{5} - \frac{3x-7}{2}\right] = 20 \left[\frac{7x+1}{4}\right] \] Distribute: - \( 20 \times \frac{2x+3}{5} = 4(2x+3) = 8x+12 \) - \( 20 \times \frac{3x-7}{2} = 10(3x-7) = 30x-70 \) - \( 20 \times \frac{7x+1}{4} = 5(7x+1) = 35x+5 \) So: \[ (8x + 12) - (30x - 70) = 35x + 5 \] --- # Step 3: Simplify the Equation Expand the left side: \[ 8x + 12 - 30x + 70 = 35x + 5 \] \[ (8x - 30x) + (12 + 70) = 35x + 5 \] \[ -22x + 82 = 35x + 5 \] --- # Step 4: Collect Like Terms Move all terms involving \(x\) to one side and constants to the other: \[ -22x - 35x = 5 - 82 \] \[ -57x = -77 \] --- # Step 5: Solve for \(x\) \[ x = \frac{-77}{-57} = \frac{77}{57} \] Simplify fraction: - Both 77 and 57 are divisible by 11. - \(77 \div 11 = 7\) - \(57 \div 11 = 5.18\) (not an integer) Check for other common factors: - 77 = 7 × 11 - 57 = 3 × 19 No common factors except 1. So, already simplified. --- # Step 6: Final Answer \[ \boxed{\frac{77}{57}} \] # Summary Table | Step | Operation | Result | |-----------------|------------------------------------------------|------------------------| | 1 | Find LCD (5,2,4) | 20 | | 2 | Multiply both sides by 20 | \(8x+12 - 30x+70 = 35x+5\) | | 3 | Simplify | \(-22x+82=35x+5\) | | 4 | Collect like terms | \(-57x=-77\) | | 5 | Solve for \(x\) | \(x = \frac{77}{57}\) | --- # Final Solution **The solution to the equation is:** \[ \boxed{x = \frac{77}{57}} \]