To determine the equation of the second street in standard form, let's follow these steps:
1. Identify the given first street equation:
The first street's equation is given as [tex]\(x + y = 6\)[/tex].
2. Find the slope of the first street:
Convert the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex] to find the slope [tex]\(m\)[/tex].
Starting with [tex]\(x + y = 6\)[/tex]:
- Subtract [tex]\(x\)[/tex] from both sides: [tex]\(y = -x + 6\)[/tex].
Therefore, the slope ([tex]\(m\)[/tex]) is [tex]\(-1\)[/tex].
3. Understand that parallel lines have the same slope:
Since the streets are parallel, the second street will also have a slope of [tex]\(-1\)[/tex].
4. Use the point-slope form to construct the equation of the second street:
The point-slope form of a line's equation is [tex]\(y - y_1 = m(x - x_1)\)[/tex] where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope.
Given the point [tex]\((1, 5)\)[/tex] and the slope [tex]\(-1\)[/tex], we substitute these values into the point-slope form:
[tex]\[
y - 5 = -1(x - 1)
\][/tex]
5. Simplify to get the line equation:
Distribute and rearrange the terms:
[tex]\[
y - 5 = -x + 1
\][/tex]
Add 5 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -x + 6
\][/tex]
6. Convert to standard form:
The goal is to get the format [tex]\(Ax + By = C\)[/tex], where [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are integers.
Adding [tex]\(x\)[/tex] to both sides of [tex]\(y = -x + 6\)[/tex]:
[tex]\[
x + y = 6
\][/tex]
Therefore, the equation of the second street in standard form is [tex]\(x + y = 6\)[/tex].
7. Identify the correct answer from the given options:
- [tex]\(x + y = 6\)[/tex]
- [tex]\(2x + y = 7\)[/tex]
- [tex]\(x - y = 6\)[/tex]
- [tex]\(2x - y = 7\)[/tex]
The correct answer is [tex]\(\boxed{x + y = 6}\)[/tex].
