To determine the equation of a line that is parallel to the street passing through the points [tex]\((4,5)\)[/tex] and [tex]\((3,2)\)[/tex] and also passes through the point [tex]\((2,-3)\)[/tex], we need to follow these steps:
1. Find the slope of the given street:
The slope (m) of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points [tex]\((4,5)\)[/tex] and [tex]\((3,2)\)[/tex]:
[tex]\[
m = \frac{2 - 5}{3 - 4} = \frac{-3}{-1} = 3
\][/tex]
Therefore, the slope of the line passing through [tex]\((4,5)\)[/tex] and [tex]\((3,2)\)[/tex] is 3.
2. Identify the slope of the parallel line:
Parallel lines have the same slope. Therefore, the slope of the parallel line passing through the point [tex]\((2,-3)\)[/tex] is also 3.
3. Determine the y-intercept (b) of the parallel line:
Use the slope-intercept form of the linear equation, which is:
[tex]\[
y = mx + b
\][/tex]
Substitute the slope (m = 3) and the coordinates of the point [tex]\((2, -3)\)[/tex] into the equation to find the y-intercept ([tex]\(b\)[/tex]):
[tex]\[
-3 = 3(2) + b
\][/tex]
Simplify the equation:
[tex]\[
-3 = 6 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -3 - 6
\][/tex]
[tex]\[
b = -9
\][/tex]
4. Write the equation of the parallel street:
Now that we have the slope ([tex]\(m = 3\)[/tex]) and the y-intercept ([tex]\(b = -9\)[/tex]), we can write the equation of the parallel street in slope-intercept form:
[tex]\[
y = 3x - 9
\][/tex]
Thus, the equation of the parallel street that passes through the point [tex]\((2, -3)\)[/tex] is:
[tex]\[
\boxed{y = 3x - 9}
\][/tex]
