Coordinate Plane

Question 1

Line a is perpendicular to line b. If the slope of line a is a positive integer, then the product of the slopes of lines a and b cannot equal

(a) -5
(b) -1
(c) 1

Answer 1

(c) 1

Lines a and b are perpendicular which means that their slopes are the opposite reciprocals of each other. Therefore, their product cannot be a positive number.

Question 2

Which point is not going to be on the line described by the following equation y = kx + k?

(a) (1, 2k)
(b) (0, k)
(c) (2, 3k)
(d) (0, -k)

Answer 2

(d) (0, -k)

Realize that you are given the x and y coordinates of four points in question and the linear equation in the slope-intercept form. Plug in the coordinates of each point into the equation and find the point whose coordinates prove the equation false.

For example, plugging in choice (c) you get 3k = k * 2 + k; 3k = 3k which is true. Choice (d) results in -k = k which is false.

Question 3

If point A with coordinates (x, y) is in quadrant III, then point B with coordinates (-x, -y) is in quadrant …?

(a) Q I
(b) Q II
(c) Q III
(d) Q IV

Answer 3

(a) Q I

Since point A is located in Q III, both x and y are negative. Negative of a negative is positive therefore, the coordinates of point B are both positive. In Q I both x and y coordinates are positive.

Question 5

Write the equation of the line that is parallel to the line y = 36x - 12 and passes through the origin.

Answer 5

y = 36x

If the line passes through the origin, its y-intercept is zero. Two lines are parallel, therefore, their slopes are equal.

Question 6

Line m is perpendicular to line p. If the slope of line p is a non-zero integer, then the slope of p divided by the slope of m could equal

(a) ¹/₉
(b) 9
(c) -¹/₉
(d) -9

Answer 6

(d) -9

Lines m and p are perpendicular which means that their slopes are the opposite reciprocals of each other. Therefore, their quotient cannot be a positive number. Eliminate choices (a) and (b).

Since the slope of line p is an integer, the slope of line m is a fraction. Dividing by a fraction is the same as multiplying by an integer. So, the right answer is an integer. The only choice that satisfies all conditions is choice (d).

Question 8

There are four parallel lines. Each line has an integer slope. The product of their slopes could be…?

(a) negative
(b) prime
(c) 0
(d) ¹/₈

Answer 8

The product of their slopes could be 0, choice (c).

Parallel lines have equal slopes. The slopes of each of four lines can be positive or negative integers but their product will always be positive. Eliminate (a). The product of slopes cannot be a prime number since there are four factors. Eliminate (b). The product of four integers cannot be a fraction. Eliminate (d).

Question 10

If the coordinates of the vertices of a rectangle are

A(2, 0), B(2, 4), C(5, 4), D(5, 0),

what is the distance between the vertices that lay on its diagonal?

Answer 10

The distance between two points on a diagonal of the rectangle ABCD is 5.

The diagonal vertices are A and C or B and D. Use distance formula to answer the question without graphing the rectangle.

Working with points A and C, you get (4 - 0)² + (5 - 2)² = 25. The square root of 25 equals 5. d = 5.

Question 11

Find the midpoint between two points with the coordinates (5, 7) and (-3, -5).

Answer 11

The coordinates of the midpoint are (1,1).

You don’t have to memorize the formula as long as you know the concept that finding the midpoint is like finding the average of the x and the y coordinates of the two points in question.

So, ^{5 + (-3)}/₂ = 1

^{7 + (-5)}/₂ = 1

Question 13

If A(4, 6) is the midpoint of DC with D = (1, 2) and C = (x, y), what are the values of x and y?

Answer 13

The value of x is 7, the value of y is 10.

Since A is the midpoint of DC,

^{(1 + <em>x</em>)}/₂ = 4

^{(2 + <em>y</em>)}/₂ = 6

Now solve for x and y.

Question 16

Without graphing, find the x and y intercepts of the line described by the following equation.

y = -2x + 2

Answer 16

• The x-intercept of this line is 1
• The y-intercept of this line is 2
• y* equals zero at the x-intercept.
• 2x + 2 = 0 ⇒ x = 1
• x* equals zero at the y-intercept.
• y* = 2

Question 17

What is a coordinate plane used for?

Answer 17

A coordinate plane is used to graph ordered pairs, straight lines and functions.

Question 18

The lines below have the same graph.

y = 4x - 6

¹/₂y = mx + b

Find the product of m and b.

m * b = ?

Answer 18

m * b = -6

The lines with the same graph have equal slopes and y-intercepts.

You can either divide the first equation by 2 or multiply the second equation by 2 to compare them.

• y* = 4x - 6
• y* = 2mx + 2b

Since the slopes and the y-intercepts are equal, conclude that 2m equals 4 and 2b equals -6.

• m* = 2; b = -3
• m * b* = -6

Question 19

What is the definition of an ordered pair?

Answer 19

An ordered pair is two numbers written in a specific order.

It is used to locate a point on a coordinate plane.

Usually denoted (x,y) where x and y are the coordinates of the point.

Example:

A (3, -8)

B (-6, -2)

Question 20

How do you plot a point on a coordinate plane?

Answer 20

To plot a point, you need to know the x and y coordinates of that point (the ordered pair).

• a point with (0,0) coordinates is the origin
• a point with (0,y) is the point on y-axis
• a point with (x,0) is the point on x-axis

Question 21

The graph of y = 5 is perpendicular to the graph of?

(a) y = 0
(b) y = x + 5
(c) x = 5

Answer 21

(c) x = 5

Question 22

There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.

In which quadrant are both x and y coordinates of a point

• positive?
• negative?

Answer 22

• The x and y coordinates of a point are both positive in the first quadrant (QI)
• The x and y coordinates of a point are both negative in the third quadrant (QIII)
• x* > 0; y > 0 - first quadrant
• x* < 0; y < 0 - third quadrant

Question 23

There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.

In which quadrant(s) do the x and y coordinates of a point have different signs?

Answer 23

The x and y coordinates of a point have different signs in the second and the fourth quadrants (Q II and Q IV).

• x* < 0; y > 0 - second quadrant
• x* > 0; y < 0 - fourth quadrant

Question 24

In which quadrant(s) lie the points with the coordinates x and y such as xy < 0?

Answer 24

Those points lie in the second and the fourth quadrants (Q II and Q IV).

The question is asking you to find the points with different sign coordinates since the product of x and y must be negative.

• x* < 0; y > 0 - second quadrant
• x* > 0; y < 0 - fourth quadrant

Question 25

How do you find the slope of a line when given the coordinates of two points on that line?

point A (3;4)

point B (2;1)

Answer 25

The slope of a line is the rise of that line over the run.

The rise: y_B - y_A = 1 - 4 = -3

The run: x_B - x_A = 2 - 3 = -1

The slope is ^-3/_-1 = 3.

Question 26

What is true about the slopes of two perpendicular lines? (excluding vertical and horizontal lines)

(a) their quotient equals 1
(b) their product equals -1
(c) their absolute values are equal
(d) none of the above
(e) all of the above

Answer 26

(b) The product of the slopes of two perpendicular lines equals -1.

The slopes of perpendicular lines are the opposite reciprocal of each other.

Question 27

What is true about the slopes of two parallel lines? (excluding vertical and horizontal lines)

(a) their quotient equals 1
(b) their product equals -1
(c) their absolute values are equal
(d) both (a) and (c)
(e) all of the above

Answer 27

(d) both (a) and (c)

The slopes of parallel lines are equal so their quotient equals 1. Naturally, the same numbers have the same absolute values.

Question 28

If given an equation of a line, how do you find its slope?

ax + ky = c

Answer 28

To find the slope of a line from an equation, put the equation into the slope-intercept form.

y = mx + b where m is the slope

• ax + ky = c*
• ky = c - ax*

y = ^c/_k - ^a/_kx

This is the slope-intercept form of the standard form of the equation. The slope equals - ^a/_k.

Question 29

Which of the following lines is parallel to the line y = 7 - 5x?

• I. y = 7 + ¹/₅x
• II. y = -7 - 5x
• III. y = 7 + 5x

(a) I only
(b) II only
(c) III only
(d) none of the above

Answer 29

(b) II only

The slopes of parallel lines are the same. Only II has the same slope as the line given in the question stem.

Question 30

At what point does the line described by the equation below cross y-axis?

10x - 5y = 25

(a) (5, 0)
(b) (0, 5)
(c) (25, 5)
(d) (0, -5)
(e) (0, 10)

Answer 30

(d) (0, -5)

You need to find the y-intercept of the line so put the equation in the slope-intercept form.

10x - 25 = 5y ⇒ 2x - 5 = y

Now, it’s clear that the y-intercept of this line has the coordinates (0, -5).

*** The x-coordinate of y-intercept is always zero so eliminate (a) and (c) right away.

Question 31

How do you find the slope of a line when given the standard form of linear equation for that line?

3x + 6y = 12

Answer 31

Rewrite this equation in the slope-intercept form. In other words, isolate the y variable.

y = mx + b

3x + 6y = 12 ⇒ 6y = 12 - 3x

y = 2 - ¹/₂x = - ¹/₂x + 2

The slope is -¹/₂.

Question 32

Find the y-intercept of the line described by the equation below.

8x + 2y = 16

Answer 32

The y-intercept is the point with (0, 8) coordinates.

• By definition, the y-intercept of a line has (0, y) coordinates. Plug in 0 into the equation and solve for y.

2y = 16 ⇒ y = 8

• Or you can rewrite the equation in the slope-intercept form.

y = 4x + 8

Question 33

What type of graph represents a linear equation?

Answer 33

A linear equation is an equation whose graph is a straight line.

Question 34

Write a standard form of a linear equation.

Answer 34

ax + by = c

where a, b and c are integers and both a and b cannot equal zero.

Question 35

What is y-intercept?

Answer 35

y-intercept of a line is the point where this line crosses the y-axis.

Question 36

What equation represents slope-intercept form of a linear equation?

Answer 36

y = mx + b

• m* stands for the slope,
• b* is y-intercept.

Question 37

How can you prove that b is the y-intercept of a line in the equation below?

y = mx + b

Answer 37

By definitions, y-intercept of a line is the point where this line crosses y-axis. Therefore, the x coordinate of that point is zero. Plug in zero in the equation below and you’ll see that y equals b.

y = mx + b

y = m * 0 + b ⇒ y = b

Question 38

Define:

The slope of a line.

Answer 38

The slope (m) of a line measures the steepness of that line.

• y = mx + b*
• m* is a ratio that shows the change in value of y to the change of value of x.
• m* =

Question 39

When is the slope of a line:

• negative?
• positive?

Answer 39

• When the slope of a line is negative, the line goes down left to right.
• When the slope is positive, the line rises up left to right.

Question 40

Fill in the blanks:

If two lines have the same graph,

• their _____ are ______

and

• their _____ are ______ .

Answer 40

If two lines have the same graph, their slopes are equal and their y-intercepts are equal.

Question 41

When two lines are parallel, their slopes are….?

Answer 41

Parallel lines have equal slopes.

Question 42

When two lines are perpendicular to each other, their slopes are….?

Answer 42

Perpendicular lines have the slopes that are opposite reciprocals of each other.

The product of the slopes of perpendicular lines equals -1.

m₁ x m₂ = -1

*** m₁ and m₂ - slopes of l₁ and l₂

l₁ and l₂ are perpendicular, nonvertical, nonhorizontal straight lines.

Question 46

What is the definition of a coordinate plane?

Answer 46

A coordinate plane is a plane formed by the intersection of a vertical (y-axis) and a horizontal (x-axis) number lines.