Distance Between Two Parallel Lines y=x+4 and -x+y=-5

Question:

Line $\ell_{1}$ has the equation $y=x+4$ and line $\ell_{2}$ has the equation $-x+y=-5$. Find the distance between $\ell_{1}$ and $\ell_{2}$. Round your answer to the nearest tenth.

Submit

The distance between the two lines is approximately 0.7 units.

Work it out

Method: Hints

To find the distance between two parallel lines, we can use the formula for the distance from a point to a line and apply it to any point on one line and the other line.

Calculation Steps and Description

1. Rearrange Equations: First, we need to write both equations in the standard form $Ax + By+C = 0$.

   • For $\ell_{1}$: $y = x + 4$ becomes $-x + y +(-4)= 0$.
   • For $\ell_{2}$: $-x + y = -5$ becomes $-x + y +(5)= 0$.

2. Identify A, B, and C: From the standard form, identify the coefficients A, B, and C for both lines.

   • For $\ell_{1}$: $A_1 = -1$, $B_1 = 1$, $C_1 = -4$.
   • For $\ell_{2}$: $A_2 = -1$, $B_2 = 1$, $C_2 = 5$.

3. Check Parallelism: Ensure that the lines are parallel by confirming that $A_1/B_1 = A_2/B_2$.

   • $(-1)/1 = (-1)/1$, so the lines are parallel.

4. Distance Formula: Use the distance formula for parallel lines: $D = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}$.

   • Here, $A$ and $B$ are the coefficients from either line (since they are the same for both lines).

5. Plug in Values: Substitute the values of $A$, $B$, $C_1$, and $C_2$ into the formula.

   • $D = \frac{|5 + (-4)|}{\sqrt{(-1)^2 + 1^2}}$.

6. Calculate: Simplify the expression to find the distance.

   • $D = \frac{|1|}{\sqrt{1 + 1}} = \frac{1}{\sqrt{2}}$.

7. Simplify: Calculate the exact distance and then round to the nearest tenth.

   • $D \approx \frac{1}{1.414} \approx 0.70721$.
   • Rounded to the nearest tenth: $D \approx 0.7$ units.

Check the Answer

After calculating the distance, we recheck the formula and the values plugged in to ensure there are no errors.

Related Knowledge Points and Detailed Explanation

• The distance between two parallel lines is constant.
• The standard form of a line is useful for applying the distance formula.
• The distance formula for a point to a line can be adapted for parallel lines by considering any point on one line and the equation of the other line.

Similar Question

A similar question might involve finding the distance between two non-parallel lines, which would require finding the point of intersection and then using the distance formula between two points.