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: Reflect the line 2x - 7y + 5 = 0 against the y-axis. When a point (x,y) is reflected across the y-axis, its new coordinates (x',y') are given by x' = -x and y' = y. From these relations, we can express x and y in terms of x' and y': x = -x' y = y' Substitute these into the original line equation: 2(-x') - 7(y') + 5 = 0 -2x' - 7y' + 5 = 0 Multiply by -1 to get a positive coefficient for x: 2x' + 7y' - 5 = 0 So, the equation of the line after reflection is 2x + 7y - 5 = 0. Step 2: Translate the reflected line 2x + 7y - 5 = 0 by the translation vector T = -8 \\ 4 . When a point (x',y') is translated by T = a \\ b , its new coordinates (x'',y'') are given by x'' = x' + a and y'' = y' + b. Here, a = -8 and b = 4. So, x'' = x' - 8 x' = x'' + 8 And y'' = y' + 4 y' = y'' - 4 Substitute these into the equation of the reflected line: 2(x'' + 8) + 7(y'' - 4) - 5 = 0 2x'' + 16 + 7y'' - 28 - 5 = 0 2x'' + 7y'' + (16 - 28 - 5) = 0 2x'' + 7y'' - 17 = 0 The final equation of the line after translation is 2x + 7y - 17 = 0. The correct option is 2x+7y-17=0. Got more? Send 'em.