To determine the equation of a street parallel to a given one, we first need to find the slope of the original street. A line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] has a slope [tex]\(m\)[/tex] which can be calculated by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
In this case, the given points on the original line are [tex]\((6,4)\)[/tex] and [tex]\((5,2)\)[/tex]. So we can calculate the slope as follows:
[tex]\[ m = \frac{2 - 4}{5 - 6} = \frac{-2}{-1} = 2 \][/tex]
Now that we have the slope [tex]\(m = 2\)[/tex], we know that any parallel line will have the same slope.
Next, we need to find the equation of the line passing through the point [tex]\((-2, 6)\)[/tex] with the same slope. We use the point-slope form of the line equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\(m = 2\)[/tex] and the point [tex]\((-2, 6)\)[/tex]:
[tex]\[ y - 6 = 2(x + 2) \][/tex]
We simplify this to find the equation in slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[ y - 6 = 2x + 4 \][/tex]
[tex]\[ y = 2x + 4 + 6 \][/tex]
[tex]\[ y = 2x + 10 \][/tex]
So, the equation of the parallel street that passes through [tex]\((-2, 6)\)[/tex] is:
[tex]\[ y = 2x + 10 \][/tex]
Therefore, the correct answer is:
[tex]\[ y = 2x + 10 \][/tex]
