Algebra I

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A straight line passes through the points and .

What is the -intercept of this line?

Explanation

First calculate the slope:

$(\text{change in Y})/(\text{change in x}) = (1-2)/(3-2) = -1$

The standard equation for a line is .

In this equation, is the slope of the line, and is the -intercept. All points on the line must fit this equation. Plug in either point (1,3) or (2,2).

Plugging in (1,3) we get .

Therefore, .

Our equation for the line is now:

To find the -intercept, we plug in :

Thus, the -intercept the point (4,0).

What line is perpendicular to $y=\frac{1}{2}x+3$ through ?

Explanation

Perdendicular lines have slopes that are opposite reciprocals. The slope of the old line is $\frac{1}{2}$ , so the new slope is .

Plug the new slope and the given point into the slope intercept equation to calculate the intercept:

or , so .

Thus , or .

Which of the following pairs of lines are parallel?

$y=\frac{1}{2}x-8$

Explanation

Lines can be written in the slope-intercept form:

In this form, equals the slope and represents where the line intersects the y-axis.

Parallel lines have the same slope: .

Only one choice contains tow lines with the same slope.

The slope for both lines in this pair is .

Which of the following lines is parallel to

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

$y = \frac{1}{2}x - 5$

Explanation

When comparing two lines to see if they are parallel, they must have the same slope. To find the slope of a line, we write it in slope-intercept form

where m is the slope.

The original equation

will need to be written in slope-intercept form. To do that, we will divide each term by 4

$\frac{4y}{4} = \frac{-2x}{4} + \frac{8}{4}$

$y = \frac{-1x}{2} + 2$

$y = - \frac{1}{2}x + 2$

Therefore, the slope of the original line is $-\frac{1}{2}$ . A line that is parallel to this line will also have a slope of $-\frac{1}{2}$ .

Therefore, the line

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

is parallel to the original line.

Find a line parallel to the line that has the equation:

$y=-\frac{1}{20}x-2$

Explanation

Lines can be written using the slope-intercept equation format:

Lines that are parallel have the same slope.

The given line has a slope of:

Only one of the choices also has the same slope and is the correct answer:

What line is perpendicular to $y=\frac{1}{2}x+3$ through ?

Explanation

Perdendicular lines have slopes that are opposite reciprocals. The slope of the old line is $\frac{1}{2}$ , so the new slope is .

Plug the new slope and the given point into the slope intercept equation to calculate the intercept:

or , so .

Thus , or .

Write an equation in slope-intercept form for the line that passes through $\left ( 3,1 \right )$ and that is perpendicular to a line which passes through the two points $\left ( 3,1 \right )$ and $\left ( -2,5 \right )$ .

$y=\frac{5}{4}X-\frac{11}{4}$

$y=-\frac{4}{5}X+11$

$y=\frac{5}{4}X+\frac{11}{4}$

Explanation

Find the slope of the line through the two points. It is $-\frac{4}{5}$ .

Since the slope of a perpendicular line is the negative reciprocal of the original line, the new line's slope is $\frac{5}{4}$ . Plug the slope and one of the points into the point-slope formula $y-y_{1}=m(x-x_{1})$ . Isolate for .