Graphing Linear Equations

Graphing linear equations means using a graph and showing the linear equation on it. This helps to find the linear relationships by plotting a graph and also can be used to find the solution to a system of linear equations.

Rectangular coordinate system:

If two lines are perpendicular to each other and positioned horizontally and vertically, then you have a rectangular coordinate system which is also called a Cartesian coordinate system. The horizontal line is called the X-axis and the vertical line is called the Y-axis. The other names for X-axis and Y-axis are abscissa and ordinate respectively. The figure shows X-axis and Y-axis. While there are four quadrants I, II, III, and IV as shown in the figure. Note: the name given to the quadrants helps to understand the system better and so should be remembered as shown in the figure.

From the figure, it is also clear that an infinite amount of values and points can be placed on both axes. The point where both lines intersect is called the origin. It divides each axis into positive and negative sections. Notice that the right-hand side of the X-axis or the horizontal axis has all the positive numbers which start from zero called the origin. The left part shows the negative numbers. Similarly, on the Y axis or the vertical axis the values above the origin are positive while below the origin are negative.

Quadrants:

Quadrant I: In the first quadrant, the value of X and the value of Y are positive. If you want to plot a number (2,2) it will always lie in the first quadrant.

Quadrant II: In the second quadrant, the value of X is negative while the value of Y is positive. So a number(-2,2) will always lie in the second quadrant.

Quadrant III: (-2,-2) falls in the 3rd quadrant. This is because the value of x which is -2 and the value of Y which is again -2 are negative. So if the value of X is negative and Y is negative, the point lies in III Quadrant.

Quadrant IV: From the figure, we can see the value of X is positive and the value of Y is negative in the 4th quadrant. So, a point (2,- 2) lies in the IV Quadrant.

Ordered Pairs:

A rectangular coordinate system can be used to show the location of one or more points. A point is represented on a graph as a solid dot (or a very small black circle). The location of the point on the graph depends on its assigned x and y values that are often expressed inside parentheses like
this: (x, y). A point that is expressed in this manner is called an ordered pair (the order is always the x-value first, then a comma, followed by the y-value next). The x and y values are called the coordinates of the point. The x-value tells us where to go along the x-axis and the y-value tells us
where to go along the y-axis. Where the x and y values intersect (or meet) is the location of the point that is being specified.

Graphing Ordered Pairs:

To identify or plot (or graph) points on the Cartesian plane (or coordinate system) you need two numbers that comprise a coordinate pair (an x and y value) that corresponds to each point. By inspecting the signs of the coordinate pair or by the location of the point on the graph, it is easy to identify the quadrant that any given point lies in. Let´s plot the following four points on the graph below:

Notice, that a point (shown as a small black dot) has been placed for point A, given by the coordinates (2, 5) at the intersection of the vertical light-colored line going through 2 on the x-axis and the horizontal line going through 5 on the y-axis. The same is true for the positioning of the other
three points labeled B, C, and D

Linear Equations:

Up until now, you have mostly seen expressions and equations with only one variable, such as:

x^2 + 17x – 5 or 19t + 5 = 12

However, now we are going to consider equations that have two different variables, such as:

y = 3x + 1 or y = -6x – 5

Notice in the examples above that for the equation y = 3x + 1 that if we set x equal to 1 (or x = 1), we can solve the equation for y by substituting 1 for x. Doing this substitution, we have

y = 3(1) + 1

or y = 4. When we have a pair of x and y values such as x = 1 and y = 4 that satisfy the equation (or make it true), this is called a solution that can be written as an ordered pair, (1, 4). Remember that the order of the x and y values in an ordered pair is always the x-value listed first, followed
by a comma, and then the y-value next, or (x-value, y-value). Sometimes an ordered pair is also referred to as a coordinate point or simply as a coordinate or as a point.

There are an infinite number of solutions to the equation y = 3x + 1. As an example of another solution, let us set x equal to 2 (or x = 2). Substituting the x with 2, we can solve for y,

y = 3(2) + 1

or y = 7. So that the ordered pair (2, 7) is another solution. Still, other solutions include (3, 10), or even (-2, -5) since when x = -2,

we obtain y = 3(-2) + 1 = -6 + 1 = -5.

Plotting Linear Equations:

These points four points (1, 4), (2,7), (3, 10), (-2,- 5) are shown on the graph. Notice that each of these coordinate pairs lie directly on the line since they are solutions to y = 3x + 1. What we have just graphed are points that are solutions to a linear equation. Linear equations are equations where if you were to graph all the ordered pairs that are solutions to the equation, they would lie along a straight line, and the line would extend into infinity
because there are an infinite number of solutions as x increases to infinity or as x decreases to minus infinity. A linear equation can always be represented in the general form y = mx + b; where m is called the “slope” or the coefficient (that is, the number that is multiplied by the variable x) and b is a constant (the y-coordinate value where the line intersects the y-axis). Notice that the variable x in a linear equation is raised to the power of 1 (in other words x as shown in the general form is the same as x^1). We learned earlier that this can be referred to as a first-degree polynomial. An equation that contains x raised to the 2nd power, or x^2, is not linear (or is non-linear) and is referred to as a quadratic equation (or 2nd-degree polynomial) which we will address at a later time.

Finding Solutions to Linear Equations and Graphing Linear Equations:

It is straightforward to find solutions and graph linear equations. We simply substitute at least two different values for x into the equation and solve for the two corresponding values of y. Then, we plot the two points and draw a line through them.

Let’s demonstrate the procedure with the linear equation,

y = -2x – 1

Usually, x = 0 is an easy substitution to use so that we have

y = -2(0) – 1 = 0 – 1 = -1

So, the first coordinate point we have on the line is (0, -1). Now, let’s try substituting x = 1 into our equation so that we have

y = -2(1) – 1 = -2 – 1 = -3

So, our second coordinate point is (1, -3). Even though we just need two points to define a line, let’s go ahead and determine a third coordinate point using x = -1. Then we have

y = -2(-1) – 1 = 2 – 1 = 1

So, our third coordinate point is (-1, 1). Wasn’t that easy! Now we can simply plot our points,

(0, -1), (1, -3), and optionally (-1, 1)

and draw a line through them to graphically represent the linear equation y = -2x – 1.

Note:

The expression on the right side of the equation y = -2x – 1 contains the variable x, so we can say that y is a function of x (or y is dependent on the value that is assigned to x), where the function f(x) is equal to -2x – 1. This equivalent notation is sometimes used: f(x) = -2x – 1, where
f(x) is the same as y and indicates that the expression is a function of (or depends on) x. When x = 4, we can compute that y = -9. Equivalently, f(4) can be computed by making the assignment x = 4 and evaluating the expression on the right side—so that f(4) = -2•4 – 1, or f(4)= -9. Both of these conventions are commonly used so it is important to be familiar with this alternate notation.

Graphing a System of Linear Equations:

When we are interested in two (or more) linear equations taken together at the same time, this is called a system of linear equations. Since each linear equation represents a line on the graph, two linear equations will usually intersect at some point that is common to both lines. This point of intersection is called the solution of the system of linear equations and it can be specified as a coordinate in the form (x, y). Let’s look at an example.

As the temperature increases in Chicago, ice cream sales increase, but hot dog sales decrease. This intuitively makes sense since something cold, such as ice cream, is a nice treat in the summer heat; yet, a hot snack, such as a hot dog, is not very popular in such heat.

After some research, it was discovered that the estimated number of ice cream sales per day, y, is given by the linear equation:

y = 3x – 25

where x is the temperature in degrees centigrade. Similarly, the estimated number of hot dog sales is given by the linear equation:

y = – 4x + 115

Charts For Ice-cream And Hot Dog Sales

The charts and graph below show how ice cream sales and hot dog sales vary with temperature.

We have set the x-value to 0, then 10, then 20, 30, and 40 in both equations that describe ice cream and hot dog sales and show the corresponding y-values that we obtained using the equations. Notice that at degree C (and lower negative temperatures), ice cream sales are -25—and such
a negative number of sales is not practical; also, at 30 degrees C (and higher), a negative number of hot dog sales is not practical. Both of the lines that represent sales as a function of temperature (that is, sales are dependent on temperature) are plotted below. Notice the point of intersection
of the two lines below.

Plotting Graph For Intersection Of Lines:

Example 1:

At what temperature are ice cream sales and hot dog sales the same?

Solution

The point of intersection occurs at a temperature of 20°C. Thus, if we substitute x = 20, into either the ice cream or hot dog sales equation we obtain the same number of sales in each case,

y = 3(20) – 25 = 60 – 25 = 35 or y = -4(20) + 115 = -80 + 115 = 35

The solution can be obtained by inspecting the graph above of the two equations. The two lines intersect at (20, 35), which is called the solution of the system of equations. Please note carefully that while point (10, 5) is located directly on the line representing ice cream sales, this point is not a solution to the system of equations—since the solution to the system of equations must consist of a point that lies on BOTH lines. In other words, the solution to the system of equations is usually only one point located at the intersection of both lines.

What happens if we substitute the values of x?

Now suppose we had chosen to substitute x = -1, x =0, and x =1, which represent temperatures of -1°C, 0°C, and 1°C, into the two linear equations, as we have done in earlier problems. Then for ice cream sales, using y =3x – 25 we would have the points (-1, -28), (0, -25), and (1, -22).

For hot dog sales, using y = -4x + 115, we would have the points (-1, 119), (0, 115), and (1,111). If we plotted and extended the lines we obtained from these points, this would yield the same solution (point of intersection) that we obtained, (20, 35).

Because of the large y-coordinates, however, our graphing of these lines would take a rather large sheet of paper, and it would take extra care to draw a rather lengthy y-axis and then extend the lines accurately. Notice that by labeling the axes 0, 5, 10, 15, 20, etc.―using 5-unit increments―we were able to plot the lines and find the intersection in a practical size on paper. Some problems may call for you to determine the scale to use on the x-and/or y-axes; it may not be practical to always increment the axes values by 1 unit when graphing a system of linear equations and attempting to find the point of intersection of the lines―especially if either the x- or y-coordinates of that point exceed perhaps 20.

Slope And y-intercept In Graphing Linear Equations

At this point in your introduction to graphing, you may have noticed that some linear equations increase more rapidly than others as the x-value increases. The following linear equations are plotted on graphs with the same scale on the x-axes and the same scale on the y-axes so that the two
slopes or rates of increase may be compared.

The difference in the angles of the two lines is called the slope. A horizontal line has a slope of 0; however, the slope increases as the line gets steeper and approaches vertical (the slope of a vertical line is infinity). The slope of a line is defined as the ratio of the change in y to the change in x between any two points on the line. If we have any two points on a given line, such as (𝑥1,𝑦1) and (𝑥2,𝑦2), then the formula for the slope is given by slope = (𝑦2− 𝑦1)/(𝑥2− 𝑥1) = (change in y)/(change in x) = ∆𝑦/∆𝑥, where ∆ is the Greek letter “delta” meaning “change” (or difference). Let’s select two points on the line y = 0.5x, (2, 1) and (4, 2), notice that we compute the slope using the above formula we obtain

slope = (2 − 1)/(4 − 2) = 1/2 or 0.5

Please notice that the slope is the same as the coefficient of x in our equation of the line y = 0.5x. Also, if you happen to use the coordinates in the reverse order, the slope is unchanged:

Conclusion: