Equations and Inequalities
In this section you will:
- Plot ordered pairs in a Cartesian coordinate system.
- Graph equations by plotting points.
- Graph equations with a graphing utility.
- Find -intercepts and -intercepts.
- Use the distance formula.
- Use the midpoint formula.
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in (Figure). Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.
Plotting Ordered Pairs in the Cartesian Coordinate System
An old story describes how seventeenth-century philosopher/mathematician René Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.
While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the x-axis and the vertical axis the y-axis.
The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in (Figure)
The center of the plane is the point at which the two axes cross. It is known as the origin, or pointFrom the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis; decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative infinity as shown by the arrowheads in (Figure).
Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its y-coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the formAn ordered pair is also known as a coordinate pair because it consists of x- and y-coordinates. For example, we can represent the pointin the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See (Figure).
When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.
Cartesian Coordinate System
A two-dimensional plane where the
- x-axis is the horizontal axis
- y-axis is the vertical axis
A point in the plane is defined as an ordered pair,such that x is determined by its horizontal distance from the origin and y is determined by its vertical distance from the origin.
Plotting Points in a Rectangular Coordinate System
Plot the points[latex]\left(3,3\right),[/latex]andin the plane.
To plot the pointbegin at the origin. The x-coordinate is –2, so move two units to the left. The y-coordinate is 4, so then move four units up in the positive y direction.
To plot the pointbegin again at the origin. The x-coordinate is 3, so move three units to the right. The y-coordinate is also 3, so move three units up in the positive y direction.
To plot the pointbegin again at the origin. The x-coordinate is 0. This tells us not to move in either direction along the x-axis. The y-coordinate is –3, so move three units down in the negative y direction. See the graph in (Figure).
Note that when either coordinate is zero, the point must be on an axis. If the x-coordinate is zero, the point is on the y-axis. If the y-coordinate is zero, the point is on the x-axis.
Graphing Equations by Plotting Points
We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.
Suppose we want to graph the equationWe can begin by substituting a value for x into the equation and determining the resulting value of y. Each pair of x– and y-values is an ordered pair that can be plotted. (Figure) lists values of x from –3 to 3 and the resulting values for y.
We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See (Figure). This is not true for all equations.
Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.
Given an equation, graph by plotting points.
- Make a table with one column labeled x, a second column labeled with the equation, and a third column listing the resulting ordered pairs.
- Enter x-values down the first column using positive and negative values. Selecting the x-values in numerical order will make the graphing simpler.
- Select x-values that will yield y-values with little effort, preferably ones that can be calculated mentally.
- Plot the ordered pairs.
- Connect the points if they form a line.
Graphing an Equation in Two Variables by Plotting Points
Graph the equationby plotting points.
Construct a table and graph the equation by plotting points:
[reveal-answer q=”3155135″]Show Solution[/reveal-answer][hidden-answer a=”3155135″]
Graphing Equations with a Graphing Utility
Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style=_____. The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.
For example, the equationhas been entered in the TI-84 Plus shown in (Figure)a. In (Figure)b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus showsandSee (Figure)c.
Finding x-intercepts and y-intercepts
The intercepts of a graph are points at which the graph crosses the axes. The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero.
To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation
To find the x-intercept, set
To find the y-intercept, set
We can confirm that our results make sense by observing a graph of the equation as in (Figure). Notice that the graph crosses the axes where we predicted it would.
Given an equation, find the intercepts.
- Find the x-intercept by settingand solving for
- Find the y-intercept by settingand solving for
Finding the Intercepts of the Given Equation
Find the intercepts of the equationThen sketch the graph using only the intercepts.
Setto find the x-intercept.
Setto find the y-intercept.
Plot both points, and draw a line passing through them as in (Figure).
Find the intercepts of the equation and sketch the graph:
x-intercept isy-intercept is
Using the Distance Formula
Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse. See (Figure).
The relationship of sidesandto side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,) The symbolsandindicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.
It follows that the distance formula is given as
We do not have to use the absolute value symbols in this definition because any number squared is positive.
The Distance Formula
Given endpointsandthe distance between two points is given by
Finding the Distance between Two Points
Find the distance between the pointsand
Let us first look at the graph of the two points. Connect the points to form a right triangle as in (Figure).
Then, calculate the length of d using the distance formula.
Find the distance between two points:and
Finding the Distance between Two Locations
Let’s return to the situation introduced at the beginning of this section.
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in (Figure). Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.
The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is atThe next stop is 5 blocks to the east, so it is atAfter that, she traveled 3 blocks east and 2 blocks north toLastly, she traveled 4 blocks north toWe can label these points on the grid as in (Figure).
Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.
- From her starting location to her first stop atTracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.
- Her second stop is atSo fromtoTracie drove east 4,000 feet.
- Her third stop is atThere are a number of routes fromtoWhatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let’s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.
- Tracie’s final stop is atThis is a straight drive north fromfor a total of 4,000 feet.
Next, we will add the distances listed in (Figure).
|From/To||Number of Feet Driven|
The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the pointsand
At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the pointPerhaps you have heard the saying “as the crow flies,” which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.[/hidden-answer]
Using the Midpoint Formula
When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment,andthe midpoint formula states how to find the coordinates of the midpoint
A graphical view of a midpoint is shown in (Figure). Notice that the line segments on either side of the midpoint are congruent.
Finding the Midpoint of the Line Segment
Find the midpoint of the line segment with the endpointsand
Use the formula to find the midpoint of the line segment.
Find the midpoint of the line segment with endpointsand
Finding the Center of a Circle
The diameter of a circle has endpointsandFind the center of the circle.
The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.
Access these online resources for additional instruction and practice with the Cartesian coordinate system.
- We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis. See (Figure).
- An equation can be graphed in the plane by creating a table of values and plotting points. See (Figure).
- Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y=_____. See (Figure).
- Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the axes. See (Figure).
- The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See (Figure) and (Figure).
- The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See (Figure) and (Figure).
Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.
Answers may vary. Yes. It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants.
Describe the process for finding the x-intercept and the y-intercept of a graph algebraically.
Describe in your own words what the y-intercept of a graph is.
The y-intercept is the point where the graph crosses the y-axis.
When using the distance formulaexplain the correct order of operations that are to be performed to obtain the correct answer.
For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.
The x-intercept isand the y-intercept is
The x-intercept isand the y-intercept is
The x-intercept isand the y-intercept is
For each of the following exercises, solve the equation for y in terms of x.
For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.
For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.
For each of the following exercises, identify the information requested.
What are the coordinates of the origin?
If a point is located on the y-axis, what is the x-coordinate?
If a point is located on the x-axis, what is the y-coordinate?
For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
[hidden-answer a=”fs-id1319659″]not collinear[/hidden-answer]
Name the coordinates of the points graphed.
Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.
For each of the following exercises, construct a table and graph the equation by plotting at least three points.
[reveal-answer q=”1538490″]Show Solution[/reveal-answer][hidden-answer a=”1538490″]
[reveal-answer q=”1764492″]Show Solution[/reveal-answer][hidden-answer a=”1764492″]
For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.
For each of the following exercises, use the graph in the figure below.
Find the distance between the two endpoints using the distance formula. Round to three decimal places.
Find the coordinates of the midpoint of the line segment connecting the two points.
Find the distance thatis from the origin.
Find the distance thatis from the origin. Round to three decimal places.
Which point is closer to the origin?
For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.
For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 2:zero button, hit enter. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit enter. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit enter. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.
Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.
Round your answer to the nearest thousandth.
A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?
If the road was made in the previous exercise, how much shorter would the man’s one-way trip be every day?
Given these four points:find the coordinates of the midpoint of line segmentsand
After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.
Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.
In the previous exercise, find the coordinates of the midpoint for each diagonal.
Midpoint of each diagonal is the same pointNote this is a characteristic of rectangles, but not other quadrilaterals.
The coordinates on a map for San Francisco areand those for Sacramento areNote that coordinates represent miles. Find the distance between the cities to the nearest mile.
If San Jose’s coordinates arewhere the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.
A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble wereOne rescue boat is at the coordinatesand a second Coast Guard craft is at coordinatesAssuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?
A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?
If we rent a truck and pay a $75/day fee plus $.20 for every mile we travel, write a linear equation that would express the total costusingto represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.
- Cartesian coordinate system
- a grid system designed with perpendicular axes invented by René Descartes
- distance formula
- a formula that can be used to find the length of a line segment if the endpoints are known
- equation in two variables
- a mathematical statement, typically written in x and y, in which two expressions are equal
- graph in two variables
- the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane
- the points at which the graph of an equation crosses the x-axis and the y-axis
- midpoint formula
- a formula to find the point that divides a line segment into two parts of equal length
- ordered pair
- a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair,
- the point where the two axes cross in the center of the plane, described by the ordered pair
- one quarter of the coordinate plane, created when the axes divide the plane into four sections
- the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right
- the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin
- the point where a graph intersects the x-axis; an ordered pair with a y-coordinate of zero
- the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top
- the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin
- a point where a graph intercepts the y-axis; an ordered pair with an x-coordinate of zero