Question

option #2 - extend and apply the skills

1. given the following equation in point - slope form y + 5 = 1/4(x + 6), convert the equation into slope - intercept form.
2. write the equation of a line perpendicular to y = x and goes through (0,0).
3. critique the reasoning of the following worked out problem. determine if it’s correct or incorrect. fix the errors if necessary.

problem: find a line parallel to y = -1/2x + 7 and goes through the point (-2,5).
student work:
y - 5 = -1/2(x + 2)

1. given the equation y = 5/7x + 3, write the equation of 3 additional lines such that the 4 lines together form a rectangle.

Explanation:

Step1: Expand the right - hand side

Expand $\frac{1}{4}(x + 6)$ in the equation $y+5=\frac{1}{4}(x + 6)$ using the distributive property $a(b + c)=ab+ac$. So, $\frac{1}{4}(x + 6)=\frac{1}{4}x+\frac{6}{4}=\frac{1}{4}x+\frac{3}{2}$. The equation becomes $y + 5=\frac{1}{4}x+\frac{3}{2}$.

Step2: Isolate y

Subtract 5 from both sides of the equation. $y=\frac{1}{4}x+\frac{3}{2}-5$. Since $5=\frac{10}{2}$, then $y=\frac{1}{4}x+\frac{3 - 10}{2}=\frac{1}{4}x-\frac{7}{2}$.

The slope of the line $y = x$ is $m_1 = 1$. If two lines are perpendicular, the product of their slopes $m_1\times m_2=-1$. So, if $m_1 = 1$, then $m_2=-1$. The line passes through the point $(0,0)$. Using the slope - intercept form $y=mx + b$, substituting $m=-1$ and the point $(0,0)$ (where $x = 0$ and $y = 0$) to find $b$, we get $0=-1\times0 + b$, so $b = 0$.

Parallel lines have the same slope. The slope of the line $y=-\frac{1}{2}x + 7$ is $m=-\frac{1}{2}$. The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-2,5)$ and $m = -\frac{1}{2}$. Substituting these values, we get $y - 5=-\frac{1}{2}(x-(-2))=-\frac{1}{2}(x + 2)$. The student's work is correct.

For a rectangle, opposite sides are parallel and adjacent sides are perpendicular. The slope of the line $y=\frac{5}{7}x + 3$ is $m_1=\frac{5}{7}$.

Parallel lines:

A parallel line will have the same slope. Let's choose a different y - intercept. For example, $y=\frac{5}{7}x+1$ and $y=\frac{5}{7}x - 2$.

Perpendicular line:

The slope of a perpendicular line $m_2$ to a line with slope $m_1$ satisfies $m_1\times m_2=-1$. So, if $m_1=\frac{5}{7}$, then $m_2=-\frac{7}{5}$. A line with this slope can be $y=-\frac{7}{5}x$.

Answer:

$y=\frac{1}{4}x-\frac{7}{2}$