## Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

### Chapter 16. The Cartesian Coordinate Plane

In this chapter, you learn about the Cartesian coordinate plane.

**Definitions for the Plane**

The *Cartesian coordinate plane* is defined by two real number lines, one horizontal and one vertical, intersecting at right angles at their zero points (see __Figure 16.1__). The two real number lines are the *coordinate axes*. The *horizontal axis*, commonly called the *x-axis*, has positive direction to the right, and the *vertical axis*, commonly referred to as the *y-axis*, has positive direction upward. The two axes determine a plane. Their point of intersection is called the *origin*.

**Figure 16.1** Cartesian coordinate plane

**Ordered Pairs in the Plane**

In the (Cartesian) coordinate plane, you identify each point *P* in the plane by an *ordered pair* (*x*, *y*) of real numbers *x* and *y*, called its *coordinates*. The ordered pair (0, 0) names the origin. An ordered pair of numbers is written in a definite order so that one number is first and the other second. The first number is the *x-coordinate*, and the second number is the *y-coordinate* (see __Figure 16.2__). The order in the ordered pair (*x*, *y*) that corresponds to a point *P* is important. The absolute value of the first coordinate, *x*, is the perpendicular horizontal distance (right or left) of the point *P* from the *y*-axis. If *x* is positive, *P* is to the right of the *y*-axis; if *x* is negative, it is to the left of the *y*-axis. The absolute value of the second coordinate, *y*, is the perpendicular vertical distance (up or down) of the point *P* from the *x*-axis. If *y* is positive, *P* is above the *x*-axis; if *y* is negative, it is below the *x*-axis.

**Figure 16.2** Point *P* in a Cartesian coordinate plane

**Problem** Name the ordered pair of integers corresponding to point *A* in the coordinate plane shown.

**Solution**

*Step 1*. Determine the *x*-coordinate of *A*.

The point *A* is 7 units to the left of the *y*-axis, so it has *x*-coordinate –7.

*Step 2*. Determine the *y*-coordinate of *A*.

The point *A* is 4 units above the *x*-axis, so it has *y*-coordinate 4.

*Step 3*. Name the ordered pair corresponding to point *A*.

(–7, 4) is the ordered pair corresponding to point *A*.

Two ordered pairs are equal if and only if their corresponding coordinates are equal; that is, (*a*, *b*) = (*c*, *d*) if and only if *a* = *c* and *b* = *d*.

**Problem** State whether the two ordered pairs are equal. Explain your answer.

**Solution**

**a**. (2, 7), (7, 2)

*Step 1*. Check whether the corresponding coordinates are equal.

(2, 7) ≠ (7, 2) because 2 and 7 are not equal.

*Step 1*. Check whether the corresponding coordinates are equal.

*Step 1*. either because – 4 ≠ 4 or because –1 ≠ 1.

**d**. because 6 = 6 and

**Quadrants of the Plane**

The axes divide the Cartesian coordinate plane into four *quadrants*. The quadrants are numbered with Roman numerals–I, II, III, and IV–beginning in the upper right and going around counterclockwise, as shown in __Figure 16.3__.

**Figure 16.3** Quadrants in the coordinate plane

Don’t forget that the quadrants are numbered *counterclockwise*.

In quadrant I, both coordinates are positive; in quadrant II, the *x*-coordinate is negative, and the *y*-coordinate is positive; in quadrant III, both coordinates are negative; and in quadrant IV, the *x*-coordinate is positive, and the *y*-coordinate is negative. Points that have 0 as one or both of the coordinates are on the axes. If the *x*-coordinate is 0, the point lies on the *y*-axis. If the *y*-coordinate is 0, the point lies on the *x*-axis. If both coordinates of a point are 0, the point is at the origin.

**Problem** Identify the quadrant in which the point lies.

**a**. (4, – 8)

**b**. (1, 6)

**c**. (– 8, – 3)

**d**. (– 4, 2)

**Solution**

**a**. (4, – 8)

*Step 1*. Note the signs of *x* and *y*.

*x* is positive and *y* is negative.

*Step 2*. Identify the quadrant in which the point lies.

Because *x* is positive and *y* is negative, (4, – 8) lies in quadrant IV.

**b**. (1, 6)

*Step 1*. Note the signs of *x* and *y*.

*x* is positive and *y* is positive.

*Step 2*. Identify the quadrant in which the point lies.

Because *x* is positive and *y* is positive, (1, 6) lies in quadrant I.

**c**. (– 8, – 3)

*Step 1*. Note the signs of *x* and *y*.

*x* is negative and *y* is negative.

*Step 2*. Identify the quadrant in which the point lies.

Because *x* is negative and *y* is negative, (– 8, – 3) lies in quadrant III.

**d**. (– 4, 2)

*Step 1*. Note the signs of *x* and *y*.

*x* is negative and *y* is positive.

*Step 2*. Identify the quadrant in which the point lies.

Because *x* is negative and *y* is positive, (– 4, 2) lies in quadrant II.

**Finding the Distance Between Two Points in the Plane**

If you have two points in a coordinate plane, you can find the distance between them using the formula given here.

** Distance Between Two Points**

The distance *d* between two points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) in a coordinate plane is given by

To avoid careless errors when using the distance formula, enclose substituted *negative* values in parentheses.

**Problem** Find the distance between the points (– 1, 4) and (5, – 3).

**Solution**

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}.

Let and Then and *y*_{2} = – 3.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the distance.

The distance between (– 1, 4) and (5, – 3) is units.

**Finding the Midpoint Between Two Points in the Plane**

You can find the midpoint between two points using the following formula.

** Midpoint Between Two Points**

The midpoint between two points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) in a coordinate plane is the point with coordinates

When you use the midpoint formula, be sure to put plus signs, not minus signs, between the two *x* values and the two *y* values.

**Problem** Find the midpoint between (– 1, 4) and (5, – 3).

**Solution**

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}. Let and Then and *y*_{2} = – 3.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the midpoint.

The midpoint between (– 1, 4) and (5, – 3) is .

**Finding the Slope of a Line Through Two Points in the Plane**

When you have two distinct points in a coordinate plane, you can construct the line through the two points. The *slope* describes the steepness or slant (if any) of the line. To calculate the slope of a line, use the following formula.

** Slope of a Line Through Two Points**

The slope *m* of a line through two distinct points, (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), is given by

When you use the slope formula, be sure to subtract the coordinates in the same order in both the numerator and the denominator. That is, if *y*_{2} is the first term in the numerator, then *x*_{2} must be the first term in the denominator. It is also a good idea to enclose substituted *negative* values in parentheses to guard against careless errors.

From the formula, you can see that the slope is the ratio of the change in vertical coordinates (the *rise*) to the change in horizontal coordinates (the *run*). Thus, __Figure 16.4__ illustrates the rise and run for the slope of the line through points *P*_{1}(*x*_{1}, *y*_{1}) and *P*_{2}(*x*_{2}, *y*_{2}).

**Figure 16.4** Rise and run

You will find it helpful to know that lines that slant upward from left to right have positive slopes, and lines that slant downward from left to right have negative slopes. Also, horizontal lines have zero slope, but the slope for vertical lines is undefined.

**Problem** Find the slope of the line through the points shown.

**Solution**

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}.

Let and Then and *y*_{2} = – 6.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the slope.

The slope of the line that passes through the points (7, 5) and (– 4, – 6) is 1. *Note:* The line slants upward from left to right—so the slope should be positive.

**Problem** Find the slope of the line through the two points.

**a**. (– 1, 4) and (5, – 3)

**b**. (– 6, 7) and (5, 7)

**c**. (5, 8) and (5, – 3)

**Solution**

**a**. (– 1, 4) and (5, – 3)

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}.

Let and Then and *y*_{2} = – 3.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the slope.

The slope of the line through (– 1, 4) and (5, – 3) is . *Note:* If you sketch the line through these two points, you will see that it slants downward from left to right—so its slope should be negative.

**b**. (– 6, 7) and (5, 7)

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}.

Let and Then and *y*_{2} = 7.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the slope.

The slope of the line that contains (– 6, 7) and (5, 7) is 0. *Note:* If you sketch the line through these two points, you will see that it is a horizontal line—so the slope should be 0.

**c**. (5, 8) and (5, – 3)

*Step 1*. Specify (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) and identify values for *x*_{1}, *y*_{1}, *x*_{2}, and *y*_{2}.

Let and Then and *y*_{2} = – 3.

*Step 2*. Evaluate the formula for the values from step 1.

*Step 3*. State the slope.

The slope of the line that contains (5, 8) and (5, – 3) is undefined. *Note:* If you sketch the line through these two points, you will see that it is a vertical line—so the slope should be undefined.

**Slopes of Parallel and Perpendicular Lines**

It is useful to know the following:

If two lines are parallel, their slopes are equal; if two lines are perpendicular, their slopes are negative reciprocals of each other.

**Problem** Find the indicated slope.

**a**. Find the slope *m*_{1} of a line that is parallel to the line through (– 3, 4) and (– 1, – 2).

**b**. Find the slope *m*_{2} of a line that is perpendicular to the line through (–3, 4) and (– 1, – 2).

**Solution**

**a**. Find the slope *m*_{1} of a line that is parallel to the line through (– 3, 4) and (– 1, – 2).

*Step 1*. Determine a strategy.

Because two parallel lines have equal slopes, *m*_{1} will equal the slope *m* of the line through (– 3, 4) and (– 1, – 2); that is, *m*_{1} = *m*.

*Step 2*. Find *m*.

*Step 3*. Determine *m*_{1}.

*m*_{1} = *m* = – 3

**b**. Find the slope *m*_{2} of a line that is perpendicular to the line through (–3, 4) and (– 1, – 2).

*Step 1*. Determine a strategy.

Because the slopes of two perpendicular lines are negative reciprocals of each other, *m*_{2} will equal the negative reciprocal of the slope *m* of the line through (– 3, 4) and (– 1, – 2); that is, .

*Step 2*. Find *m*.

*Step 3*. Determine *m*_{2}.

**Exercise 16**

*For 1—6, indicate whether the statement is true or false*.

__1__. The intersection of the coordinate axes is the origin.

__2__. (2, 3) = (3, 2)

__3__.

__4__. The point is in quadrant II.

__5__. The point is in quadrant III.

__6__. The point (5, 0) is in quadrant I.

*For 7—14, fill in the blank to make a true statement*.

__7__. The change in *y*-coordinates between two points on a line is the __________.

__8__. The change in *x*-coordinates between two points on a line is the __________.

__9__. Lines that slant downward from left to right have __________ slopes.

__10__. Lines that slant upward from left to right have __________ slopes.

__11__. Horizontal lines have __________ slope.

__12__. The slope of a line that is parallel to a line that has slope is __________.

__13__. The slope of a line that is perpendicular to a line that has slope is __________.

__14__. The slope of a vertical line is __________.

__15__. Name the ordered pair of integers corresponding to point *K* in the following coordinate plane.

__16__. Find the distance between the points (1, 4) and (5, 7).

__17__. Find the distance between the points (– 2, 5) and (4, – 1).

__18__. Find the midpoint between the points (– 2, 5) and (4, – 1).

__19__. Find the slope of the line through (– 2, 5) and (4, – 1).

__20__. Find the slope of the line through the points shown.