Algebra - Equations of Lines

Find the slope of the line that passes through the following points:

and

$-\frac{11}{19}$

$-\frac{19}{11}$

$-\frac{12}{19}$

Explanation

Use the following formula to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Remember that points are written in the following format:

For this line,

$\text{Slope}=\frac{1-12}{0-(-19)}$

Subtracting a negative number is the same as adding a positive number.

$\text{Slope}=\frac{1-12}{0+19}$

$\text{Slope}=\frac{-11}{19}$

Simplify.

$\text{Slope}=-\frac{11}{19}$

What is the slope of the line that runs through the points and ?

Undefined

Explanation

We have been given a set of coordinate points in order to find the slope of the line that runs through it.

We can use the equation:

$\frac{y_2-y_1}{x_2-x_1}= \frac{9-1}{9-9}=\frac{8}0$

Since the resulting denominator is 0, it is determined that the slope is a vertical line (since there was no horizontal movement) with an undefined slope or 'no slope'.

Given the line 4y = 2x + 1, what is the slope of this line?

1/2

1/4

–1/4

–2

Explanation

4y = 2x + 1 becomes y = 0.5x + 0.25. We can read the coefficient of x, which is the slope of the line.

4y = 2x + 1

(4y)/4 = (2x)/4 + (1)/4

y = 0.5x + 0.25

y = mx + b, where the slope is equal to m.

The coefficient is 0.5, so the slope is 1/2.

Find the slope of the line that passes through the following points:

and

$\frac{5}{6}$

$\frac{6}{5}$

$\frac{1}{3}$

Explanation

Use the following formula to find the slope of the line:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Remember that points are written in the following format:

For this line,

$\text{Slope}=\frac{7-2}{5-(-1)}$

Subtracting a negative number is the same as adding a positive number.

$\text{Slope}=\frac{7-2}{5+1}$

$\text{Slope}=\frac{5}{6}$

Find the slope of the line that passes through the following points:

and

$-\frac{1}{3}$

$\frac{1}{3}$

$-\frac{13}{3}$

Explanation

Use the following formula to find the slope:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Remember that points are written in the following format:

Substitute using the given points:

$\text{Slope}=\frac{2-6}{7-(-5)}$

Remember that subtracting a negative number is the same as adding a positive number.

$\text{Slope}=\frac{2-6}{7+5}$

Simplify.

$\text{Slope}=-\frac{4}{12}$

Reduce.

$\text{Slope}=-\frac{1}{3}$

Find the midpoint of a line segment with the following endpoints:

Explanation

Finding the midpoint of a line segment only requires that we find the midpoint, or average of both the x and y components. In order to do that, we use the following formula:

$(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

For the points (2,6) and (12,12) plug in the numbers and solve:

$(\frac{2+12}{2},\frac{6+12}{2})=(\frac{14}{2},\frac{18}{2})=(7,9)$

This gives a final answer of

What is the midpoint of a line segment with the end points and ?

Explanation

In order to solve for the midpoint of a line segment when the end points are given, the x and y values must be averaged.

A formula for this would be:
$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

This probem may be quickly solved for by substituting in the given information.

$midpoint=(\frac{1+7}{2},\frac{2+8}{2})$

$midpoint=(\frac{8}{2},\frac{10}{2})$

$midpoint={\color{Blue} (4,5)}$

$\textup{Find the distance between the midpoint of the line segment}\ \textup{ between points } (-1,-1)\textup{ and } (2,3) \textup{ and the midpoint of the line}\\textup{ segment between points }(-1,-1)\textup{ and (2,-1).}$

$\textup{None of the above.}$

$\frac{1}{3}$

$\frac{2}{3}$

$\textup{1.5}$

$\textup{4}$

Explanation

$\textup{The midpoint of the first line segment has coordinates }\\left(\frac{-1+2}{2},\frac{-1+3}{2} \right )=\left(0.5,1 \right) \textup{and the midpoint of the second}\\textup{ line segment has coordinates }\left(\frac{-1+2}{2},\frac{-1-1}{2} \right )=\left(0.5,-1 \right ).$

$\textup{The segment that joins the two midpoints is on a vertical line, }\\textup{therefore its length is the absolute difference between the two}\\textup{midpoints' }y\textup{-coordinates }\left | 1-(-1)\right |=2.$

$\textup{Alternatively, one can draw all the points and segments and find}\\textup{the distance by exploiting the similarity of the two right triangles.}$

Find the slope of the line that is perpendicular to a line with the equation:

$-\frac{1}{2}$

$\frac{1}{2}$

Explanation

Lines can be written in the slope-intercept form:

In this equation, is the slope and is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of :