Geometry - How to find the equation of a line

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

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

$y=\frac{1}{5}x+5$

Explanation

If the y-intercept of a line is , then the -value is when is zero. Write the point:

If the -intercept of a line is , then the -value is when is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5$

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

Write the equation for the line passing through the points and

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

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

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

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

$y = \frac{1}{2} x -3$

Explanation

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{3--1}{-4-4} = \frac{4 }{-8 } =- \frac{1}{2}$

We want this equation in slope-intercept form, . We know and because we have two coordinate pairs to choose from representing an and a . We know because that represents the slope. We just need to solve for , and then we can write the equation.

We can choose either point and get the correct answer. Let's choose :

$-1 = -\frac{1}{2} (4) + b$ multiply ""

add to both sides

This means that the form is $y = -\frac{1}{2}x +1$

Find the equation of a line that goes through the points and .

$y=\frac{7}{5}x+\frac{39}{5}$

$y=-\frac{1}{6}x-\frac{6}{5}$

$y=\frac{2}{3}x+\frac{4}{9}$

$y=-\frac{2}{5}x-\frac{11}{5}$

Explanation

Recall that the slope-intercept form of a line:

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

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

$\text{Slope}=\frac{5-12}{-2-3}=\frac{7}{5}$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

$y=\frac{7}{5}x+b$

Plugging in yields the following:

$12=\frac{7}{5}(3)+b$

Solve for .

$b+\frac{21}{5}=12$

$b=\frac{39}{5}$

The equation of the line is then $y=\frac{7}{5}x+\frac{39}{5}$ .

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Refer to the above figures. To the left is the graph of the equation

What inequality is graphed at right?

$y \ge -3x + 7$

$y \le -3x + 7$

Explanation

As indicated by the solid line, the graph of the inequality at right includes the line of the equation, so the inequality graphed is either

$y \ge -3x + 7$

$y \le -3x + 7$

To determine which one, we can select a test point and substitute its coordinates in either inequality, testing whether it is true for those values. The easiest test point is ; it is not part of the solution region, so we want the inequality that it makes false. Let us select the first inequality:

$y \ge -3x + 7$

$0 \ge -3 (0) + 7$

$0 \ge 7$

makes this inequality false, so the graph of the inequality $y \ge -3x + 7$ is the one that does not include the origin. This is the correct choice. (Note that if we had selected the other inequality, we would have seen that makes it true; this would have allowed us to draw the same conclusion.)

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Refer to the above diagrams. At left is the graph of the equation

At right is the graph of the equation

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

Which of the following is a graph of the system of linear inequalities

$y \le 2x+ 8$

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

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Explanation

The graph of a linear inequality that includes either the $\le$ or $\ge$ symbol is the line of the corresponding equation along with all of the points on either side of the line. We are given both lines, so for each inequality, it remains to determine which side of each line is included. This can be done by choosing any test point on either side of the line, substituting its coordinates in the inequality, and determining whether the inequality is true or not. The easiest test point is .

$y \le 2x+ 8$

$0 \le 2(0) + 8$

$0 \le 0+ 8$

$0 \le 8$

This is true; select the side of this line that includes the origin.

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

$0 \le -\frac{1}{2}(0)-4$

$0 \le 0-4$

$0\le -4$

This is false; select the side of this line that does not include the origin.

The solution sets of the individual inequalities are below:

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The graph of the system is the intersection of the two sets, shown below:

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Which of the following inequalities is graphed above?

$x \ge 5$

$y \ge 5$

$\textup{None of the other choices gives the correct response}$

Explanation

The boundary line is a horizontal line which has as its -intercept; the equation of this line is .

The inequality is either or $x \ge 5$ , since the region right of the line is included. The dashed boundary indicates that equality is not allowed, so the correct inequality is .

Find the equation of a line that goes through the points and .

$y=\frac{1}{22}x+\frac{82}{11}$

$y=-\frac{18}{19}x-\frac{6}{19}$

$y=\frac{5}{18}x+\frac{1}{9}$

Explanation

Recall that the slope-intercept form of a line:

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

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

$\text{Slope}=\frac{8-7}{12-(-10)}=\frac{1}{22}$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

$y=\frac{1}{22}x+b$

Plugging in yields the following:

$8=\frac{1}{22}(12)+b$

Solve for .

$b+\frac{6}{11}=8$

$b=\frac{82}{11}$

The equation of the line is then $y=\frac{1}{22}x+\frac{82}{11}$ .

Find the equation for a line passing through the points and .

$y = \frac{1}{3} x -8$

$y = \frac{1}{3}x -6$

Explanation

To determine the equation, first find the slope:

$\frac{\Delta y}{\Delta x} = \frac{-7--1}{3-5} = \frac{-6 }{-2 } =3$

We can choose either point and get the correct answer. Let's choose :

multiply ""

subtract from both sides

This means that the form is

Find the equation of a line that goes through the points and .

$y=\frac{1}{8}x+112$

$y=-\frac{1}{4}x-111$

Explanation

Recall that the slope-intercept form of a line:

where $m=\text{slope}$ and $b=\text{y-intercept}$ .

First, find the slope of the line by using the following formula:

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

$\text{Slope}=\frac{10-2}{16-15}=8$

Next, find the y-intercept of the line by plugging in of the points into the semi-completed formula.

Plugging in yields the following:

Solve for .

The equation of the line is then .

Given two points $\left ( -3,-3 \right )$ and $\left ( -1,1 \right )$ , find the equation for the line connecting those two points in slope-intercept form.