Can someone help me on this problem?

Question

Line ℓ1 has the equation y=–x+1

and line ℓ2 has the equation x+y=–3. Find the distance between ℓ1 and ℓ2.

Round your answer to the nearest tenth.

Answer 1

If you solve equation for L2 for y, resulting in y = -x-3, it is apparent that the lines are parallel since the their slopes are the same (m = -1). To find the distance between the lines find a line that is perpendicular to them, find the respective points if intersection, then use the distance formula on those points.

Any line that has a slope of 1 will be perpendicular to both L1 and L2. Using y = x is simplest. Call y= x L3.

L1 intersect L3 can be found by substitution. -x+1 = x, 2x = 1, x= 1/2. So point of intersection is (1/2, 1/2).

L2 intersects L3 when -x-3 = x, 2x=-3, x = -3/2. Point of intersection (-3/2, -3/2).

Find the distance between those two points and you have the distance between L1 and L2.

Check your work here:

desmos.com/calculator/mdxopmcoji

The above graph generalizes the solution a bit, using a slider for c for the equation y = x + c. Use the slider on c to see that the distance between L1 and L2 is always the same.