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Intermediate Geometry : Coordinate Geometry

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Example Questions

Example Question #31 : How To Find The Equation Of A Line

Untitled

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

y =2x+ 8

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

$y \le 2x+ 8$

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

Possible Answers:

Untitled

Correct answer:

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 (0,0) .

$y \le 2x+ 8$

$0 \le 2(0) + 8$

$0 \le 0+ 8$

$0 \le 8$

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:

Untitled

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

Untitled

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Example Question #34 : How To Find The Equation Of A Line

Untitled

Which of the following inequalities is graphed above?

Possible Answers:

$y \ge 5$

x> 5

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

$x \ge 5$

y> 5

Correct answer:

x> 5

Explanation:

The boundary line is a horizontal line which has (5,0 ) as its -intercept; the equation of this line is x = 5 .

The inequality is either x> 5 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 x > 5 .

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Example Question #35 : How To Find The Equation Of A Line

Find the equation of a line that passes through the following points:

(5, 8) and (6, 12)

Possible Answers:

$y=\frac{17}{14}x-2$

y=-4x-8

y=4x-12

y=3x-2

$y=\frac{1}{4}x-10$

Correct answer:

y=4x-12

Explanation:

Recall that the the following is the slope-intercept form of a line:

y=mx+b

In this equation, the variables are represented by the following:

$m=\text{slope}$

$b=\text{y-intercept}$

Find the slope of the line by using the following formula:

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

In this equation, the x- and y-variables correspond to the coordinates of the given points.

$\text{Slope}=\frac{12-8}{6-5}$

$\text{Slope}=\frac{4}{1}$

$\text{Slope}=4$

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

y=4x+b

Substituting in the point (5, 8) yields the following:

8=4(5)+b

Rearrange and solve for .

b+20=8

Subtract 20 from both sides of the equation.

b+20-20=8-20

b=-12

Substitute this value of the y-intercept into our semi-complete equation to get the answer:

y=4x-12

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Example Question #63 : Expressions & Equations

What is the slope of the line that passes through the points (3,4) and (5,13) ?

Possible Answers:

-9/2

13/4

2/9

-4/3

9/2

Correct answer:

9/2

Explanation:

The slope of a line is sometimes referred to as "rise over run." This is because the formula for slope is the change in y-value (rise) divided by the change in x-value (run). Therefore, if you are given two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , the slope of their line can be found using the following formula: $slope=(y_{1}-y_{2})/(x_{1}-x_{2})$

This gives us (13-4)/(5-3)=9/2 .

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Example Question #1 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

Given points (1,2) and (-1,3) , what is the slope of the line connecting them?

Possible Answers:

$-\frac{1}{2}$

$\frac{1}{2}$

$-\frac{1}{4}$

Correct answer:

$-\frac{1}{2}$

Explanation:

Write the slope formula. Plug in the points and solve.

$m= \frac{y_2-y_1}{x_2-x_1}= \frac{3-2}{-1-1}= -\frac{1}{2}$

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Example Question #2 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

What is the slope of the line connecting the points $(2,\sqrt2)$ and (1,2) ?

Possible Answers:

$\sqrt2 -2$

$-\sqrt2+1$

$-\sqrt2-2$

$2-\sqrt2$

$-1-\sqrt2$

Correct answer:

$\sqrt2 -2$

Explanation:

Write the slope formula. Plug in the point, and simplify.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{2-\sqrt2}{1-2}=\frac{2-\sqrt2}{-1}= -(2-\sqrt2) = \sqrt2-2$

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Example Question #66 : Expressions & Equations

What is the slope of a line with an -intercept is and another -intercept of ?

Possible Answers:

$\infty$

Correct answer:

Explanation:

The -intercept is the value when y=0 .

Therefore, since the two -intercepts are and , the points are (4,0) and (6,0) .

Write the slope formula, plug in the values, and solve.

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-0}{6-4}= \frac{0}{2} =0$

The slope is zero.

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Example Question #1 : How To Find The Slope Of A Line

Given the points (4, 3) and (8, 9) , find the slope of the line.

Possible Answers:

$\frac{-3}{4}$

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{-6}{4}$

$\frac{4}{3}$

Correct answer:

$\frac{3}{2}$

Explanation:

The formula for the slope of a line is $\frac{y_2-y_1}{x_2-x_1}$ .

We then plug in the points given: $\frac{9-3}{8-4}=\frac{6}{4}$ which is then reduced to $\frac{3}{2}$ .

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Example Question #2 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

A line crosses the x-axis at (6,8) and the y-axis at (0,6) . What is the slope of this line?

Possible Answers:

None of these.

m=3

$m=-\frac{1}{3}$

m=-3

$m=\frac{1}{3}$

Correct answer:

$m=\frac{1}{3}$

Explanation:

Given the points,

$\\(x_1,y_1)=(0,6) \\(x_2,y_2)=(6,8)$ .

We compute slope (m) as follows:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-6}{6-0}=\mathbf{\frac{1}{3}}$

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Example Question #1423 : Intermediate Geometry

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

(12, 2) and (-12, 4)

Possible Answers:

$-\frac{3}{4}$

$-\frac{1}{12}$

Correct answer:

$-\frac{1}{12}$

Explanation:

Recall that the slope of a line also measures how steep the line is. Use the following equation to find the slope of the line:

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

Now, substitute in the information using the given points.

$\text{Slope}=\frac{4-2}{-12-12}$

Simplify.

$\text{Slope}=\frac{2}{-24}$

Solve.

$\text{Slope}=-\frac{1}{12}$

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Intermediate Geometry : Coordinate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #31 : How To Find The Equation Of A Line

Example Question #34 : How To Find The Equation Of A Line

Example Question #35 : How To Find The Equation Of A Line

Example Question #63 : Expressions & Equations

Example Question #1 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

Example Question #2 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

Example Question #66 : Expressions & Equations

Example Question #1 : How To Find The Slope Of A Line

Example Question #2 : Graph Proportional Relationships, Interpreting The Unit Rate As The Slope: Ccss.Math.Content.8.Ee.B.5

Example Question #1423 : Intermediate Geometry

All Intermediate Geometry Resources

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