Draw a straight line through the points. Check by graphing a third ordered pair that is a solution of the equation and verify that it lies on the line. Solution We first select any two values of x to find the associated values of y. We will use 1 and 4 for x. Next, we graph these ordered pairs and draw a straight line through the points as shown in the figure. We use arrowheads to show that the line extends infinitely far in both directions.
Any third ordered pair that satisfies the equation can be used as a check: Next, we graph these ordered pairs and pass a straight line through the points, as shown in the figure. In fact, any ordered pair of the form x, 2 is a solution of 1. Graphing the solutions yields a horizontal line as shown in Figure 7. In fact, any ordered pair of the form -3, y is a solution of 2. Graphing the solutions yields a vertical line as shown in Figure 7. Some solutions are 1, 3 , 2,3 , and 5, 3.
The solutions of an equation in two variables that are generally easiest to find are those in which either the first or second component is 0. For example, if we substitute 0 for x in the equation. Thus, a solution of Equation 1 is 0, 3. We can also find ordered pairs that are solutions of equations in two variables by assigning values to y and determining the corresponding values of x.
In particular, if we substitute 0 for y in Equation 1 , we get. We can now use the ordered pairs 0, 3 and 4, 0 to graph Equation 1. The graph is shown in Figure 7. Notice that the line crosses the x-axis at 4 and the y-axis at 3. For this reason, the number 4 is called the x-intercept of the graph, and the number 3 is called the y-intercept. This method of drawing the graph of a linear equation is called the intercept method of graphing.
Note that when we use this method of graphing a linear equation, there is no advantage in first expressing y explicitly in terms of x. If the graph intersects the axes at or near the origin, the intercept method is not satisfactory.
We must then graph an ordered pair that is a solution of the equation and whose graph is not the origin or is not too close to the origin. Choosing any other value for x,say 2, we get. Thus, 0, 0 and 2, 6 are solutions to the equation. In this section, we will study an important property of a line. We will assign a number to a line, which we call slope, that will give us a measure of the "steepness" or "direction" of the line.
It is often convenient to use a special notation to distinguish between the rectan- gular coordinates of two different points. We can designate one pair of coordinates by x 1 , y 1 read "x sub one, y sub one" , associated with a point P 1 , and a second pair of coordinates by x 2 , y 2 , associated with a second point P 2 , as shown in Figure 7. Note in Figure 7.
The ratio of the vertical change to the horizontal change is called the slope of the line containing the points P 1 and P 2. This ratio is usually designated by m. Substituting into Equation 1 yields. Note that we get the same result if we subsitute -4 and 2 for x 2 and y 2 and 3 and 5 for x 1 and y 1. Lines with various slopes are shown in Figure 7.
Slopes of the lines that go up to the right are positive Figure 7. And note Figure 7. Also note Figure 7. Consider the lines shown in Figure 7. These lines will never intersect and are called parallel lines. Now consider the lines shown in Figure 7.
In general, if two lines have slopes and m2: Let us say we know that a line goes through the point 2, 3 and has a slope of 2. If we denote any other point on the line as P x, y See Figure 7. Thus, Equation 1 is the equation of the line that goes through the point 2, 3 and has a slope of 2. In general let us say we know a line passes through a point P 1 x 1 , y 1 and has slope m.
If we denote any other point on the line as P x, y see Figure 7. Equation 2 is called the point-slope form for a linear equation. In Equation 2 , m, x 1 and y 1 are known and x and y are variables that represent the coordinates of any point on the line. Thus, whenever we know the slope of a line and a point on the line, we can find the equation of the line by using Equation 2. Solution Substitute -2 for m and 2, 4 for x 1 , y 1 in Equation 2. Now consider the equation of a line with slope m and y-intercept b as shown in Figure 7.
Substituting 0 for x 1 and b for y 1 in the point-slope form of a linear equation, we have. Equation 3 is called the slope-intercept form for a linear equation.
The slope and y-intercept can be obtained directly from an equation in this form. Such a relationship is called a direct variation. We say that the variable y varies directly as x. We know that the pressure P in a liquid varies directly as the depth d below the surface of the liquid. We can state this relationship in symbols as. In a direct variation, if we know a set of conditions on the two variables, and if we further know another value for one of the variables, we can find the value of the second variable for this new set of conditions.
In this section we will graph inequalities in two variables. For example, consider the inequality. The solutions are ordered pairs of numbers that "satisfy" the inequality. That is, a, b is a solution of the inequality if the inequality is a true statement after we substitute a for x and b for y. Solution The ordered pair 1, 1 is a solution because, when 1 is substituted for x and 1 is substituted for y, we get. On the other hand, 2, 5 is not a solution because when 2 is substituted for x and 5 is substituted for y, we obtain.
Thus, every point on or below the line is in the graph. We represent this by shading the region below the line see Figure 7. Therefore, the points are 0, 1 and 2, 0.
Plot the points in the X , Y graph Step 3: Join the points to produce a line. Videos related to Algebra. Need more help understanding graphing linear equations? Top Algebra solution manuals Get step-by-step solutions. Find step-by-step solutions for your textbook.
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Steps involved in graphing linear equations: Step 1: Find the x-intercept and y-intercept of the linear equation. Need more help understanding graphing linear equations? Send any homework question to our team of experts.
To graph a linear equation, we can use the slope and y-intercept. Locate the y-intercept on the graph and plot the point. From this point, . The solution set to any equation is the place where BOTH equations meet on the xy-plane. This meeting place is called the Point of Intersection. If you have a linear equation and a quadratic equation on the same xy-plane, there may be TWO POINTS where the graph of each equation will meet or intersect.
For example, an equation such 3x +5y = 16 is already in standard form, as a equals 3, b equals 5, and c equals Similarly, an equation such as y = 5x – 3 can be put in standard form as -5x +y = Graphing Linear Functions. If the equation for a function can be written in standard form, it can be graphed as a linear function. Suppose the . Systems of linear equations can be solved by other methods than graphing, such as substitution and elimination. When solving by elimination, some systems can be solved by addition, subtraction, and multiplication.