Find the general equation of a line, which passes through the point of intersection of the lines intersection L1 ∩ L2, where L1: - 2x + 5y = 16 and L2: -7x+3y = 8, and is perpendicular to another line: L3 : -x+2y = -1

To find the equation of a line passing through the intersection of two lines, L1 and L2, and perpendicular to a third line, L3, you can follow these steps: 1. Find the intersection point of L1 and L2. 2. Determine the slope of L3. 3. Find the negative reciprocal of the slope of L3 to get the slope of the line you're looking for. 4. Use the point-slope form of the equation of a line to find the equation of the line. Let's go through these steps one by one: 1. Find the intersection point of L1 and L2: First, we'll solve the system of equations formed by L1 and L2: L1: -2x + 5y = 16 L2: -7x + 3y = 8 We can use the method of substitution or elimination to find the intersection point. I'll use elimination here. Multiply the first equation by 7 and the second equation by 2 to make the coefficients of x in both equations equal: 7(-2x + 5y) = 7(16) 2(-7x + 3y) = 2(8) This simplifies to: -14x + 35y = 112 -14x + 6y = 16 Now, subtract the second equation from the first: (-14x + 35y) - (-14x + 6y) = 112 - 16 -14x + 35y + 14x - 6y = 96 29y = 96 y = 96 / 29 Now, we can substitute this value of y into either L1 or L2 to find the x-coordinate. Let's use L1: -2x + 5(96/29) = 16 -2x + (480/29) = 16 -2x = 16 - (480/29) -2x = (464/29) - (480/29) -2x = -16/29 x = (8/29) So, the intersection point of L1 and L2 is (8/29, 96/29). 2. Determine the slope of L3: L3: -x + 2y = -1 Rewrite it in slope-intercept form (y = mx + b), where m is the slope: 2y = x - 1 y = (1/2)x - 1/2 The slope of L3 is 1/2. 3. Find the negative reciprocal of the slope of L3: The negative reciprocal of 1/2 is -2. 4. Use the point-slope form of the equation of a line to find the equation of the line passing through the intersection point and perpendicular to L3: Using point-slope form: y - y1 = m(x - x1) y - (96/29) = -2(x - 8/29) Now, let's simplify and put it in standard form: Multiply both sides by 29 to get rid of fractions: 29y - 96 = -2(29x - 8) Distribute the -2 on the right side: 29y - 96 = -58x + 16 Add 58x to both sides and add 96 to both sides: 58x + 29y = 16 + 96 58x + 29y = 112 So, the general equation of the line passing through the intersection of L1 and L2, and perpendicular to L3, is: 58x + 29y = 112