This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Cross-multiply the fractions. Given the equation: (4r-3)/(6r+1) = (2r-1)/(3r+4) Multiply both sides by (6r+1)(3r+4) to eliminate the denominators: (4r-3)(3r+4) = (2r-1)(6r+1) Step 2: Expand both sides of the equation. Expand the left side: 4r(3r) + 4r(4) - 3(3r) - 3(4) = 12r^2 + 16r - 9r - 12 = 12r^2 + 7r - 12 Expand the right side: 2r(6r) + 2r(1) - 1(6r) - 1(1) = 12r^2 + 2r - 6r - 1 = 12r^2 - 4r - 1 Now the equation is: 12r^2 + 7r - 12 = 12r^2 - 4r - 1 Step 3: Rearrange the terms to solve for r. Subtract 12r^2 from both sides: 7r - 12 = -4r - 1 Add 4r to both sides: 7r + 4r - 12 = -1 11r - 12 = -1 Add 12 to both sides: 11r = -1 + 12 11r = 11 Divide by 11: r = (11)/(11) r = 1 Step 4: Check for excluded values. The denominators cannot be zero. 6r+1 ≠ 0 6r ≠ -1 r ≠ -(1)/(6) 3r+4 ≠ 0 3r ≠ -4 r ≠ -(4)/(3) Since r=1 is not equal to -(1)/(6) or -(4)/(3), it is a valid solution. The solution is r=1.