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.

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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.