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Write the equation of a line in slope-intercept form. Graph a straight line using its slope and y-intercept. We now wish to discuss an important concept called the slope of a line. Intuitively we can think of slope as the steepness of the line in relationship to the horizontal.
Following are graphs of several lines. Study them closely and mentally answer the questions that follow. What seems to be the relationship between the coefficient of x and the steepness Which graph would be steeper: Which graph would be steeper: If m as the value of m increases, the steepness of the line decreases and the line rises to the left and falls to the right. In other words, in an equation of the form y - mx, m controls the steepness of the line.
In mathematics we use the word slope in referring to steepness and form the following definition:. Solution We first make a table showing three sets of ordered pairs that satisfy the equation. Remember, we only need two points to determine the line but we use the third point as a check. Example 2 Sketch the graph and state the slope of. Why use values that are divisible by 3?
Compare the coefficients of x in these two equations. Again, compare the coefficients of x in the two equations. Observe that when two lines have the same slope, they are parallel. The slope from one point on a line to another is determined by the ratio of the change in y to the change in x. If you want to impress your friends, you can write where the Greek letter delta means "change in. We could also say that the change in x is 4 and the change in y is - 1.
This will result in the same line. The change in x is 1 and the change in y is 3. If an equation is in this form, m is the slope of the line and 0,b is the point at which the graph intercepts crosses the y-axis. The point 0,b is referred to as the y-intercept. If the equation of a straight line is in the slope-intercept form, it is possible to sketch its graph without making a table of values. Use the y-intercept and the slope to draw the graph, as shown in example 8.
First locate the point 0, This is one of the points on the line. The slope indicates that the changes in x is 4, so from the point 0,-2 we move four units in the positive direction parallel to the x-axis. Since the change in y is 3, we then move three units in the positive direction parallel to the y-axis. The resulting point is also on the line. Since two points determine a straight line, we then draw the graph.
Always start from the y-intercept. A common error that many students make is to confuse the y-intercept with the x-intercept the point where the line crosses the x-axis. To express the slope as a ratio we may write -3 as or.
If we write the slope as , then from the point 0,4 we move one unit in the positive direction parallel to the x-axis and then move three units in the negative direction parallel to the y-axis. Then we draw a line through this point and 0,4. Can we still find the slope and y-intercept? The answer to this question is yes. To do this, however, we must change the form of the given equation by applying the methods used in section Section dealt with solving literal equations.
You may want to review that section. Solution First we recognize that the equation is not in the slope-intercept form needed to answer the questions asked.
To obtain this form solve the given equation for y. Sketch the graph of here. Sketch the graph of the line on the grid below. These were inequalities involving only one variable. We found that in all such cases the graph was some portion of the number line. Since an equation in two variables gives a graph on the plane, it seems reasonable to assume that an inequality in two variables would graph as some portion or region of the plane.
This is in fact the case. To summarize, the following ordered pairs give a true statement. The following ordered pairs give a false statement. If one point of a half-plane is in the solution set of a linear inequality, then all points in that half-plane are in the solution set. This gives us a convenient method for graphing linear inequalities. To graph a linear inequality 1. Replace the inequality symbol with an equal sign and graph the resulting line.
Check one point that is obviously in a particular half-plane of that line to see if it is in the solution set of the inequality. If the point chosen is in the solution set, then that entire half-plane is the solution set. If the point chosen is not in the solution set, then the other half-plane is the solution set. Why do we need to check only one point?
The point 0,0 is not in the solution set, therefore the half-plane containing 0,0 is not the solution set. Since the line itself is not a part of the solution, it is shown as a dashed line and the half-plane is shaded to show the solution set.
The solution set is the half-plane above and to the right of the line. Since the point 0,0 is not in the solution set, the half-plane containing 0,0 is not in the set.
Hence, the solution is the other half-plane. Therefore, draw a solid line to show that it is part of the graph. The solution set is the line and the half-plane below and to the right of the line. Next check a point not on the line. Notice that the graph of the line contains the point 0,0 , so we cannot use it as a checkpoint. The point - 2,3 is such a point.
When the graph of the line goes through the origin, any other point on the x- or y-axis would also be a good choice. Sketch the graphs of two linear equations on the same coordinate system. Determine the common solution of the two graphs. Example 1 The pair of equations is called a system of linear equations. We have observed that each of these equations has infinitely many solutions and each will form a straight line when we graph it on the Cartesian coordinate system.
We now wish to find solutions to the system. In other words, we want all points x,y that will be on the graph of both equations. Solution We reason in this manner: In this table we let x take on the values 0, 1, and 2. We then find the values for y by using the equation. Do this before going on. In this table we let y take on the values 2, 3, and 6. We then find x by using the equation. Check these values also. The two lines intersect at the point 3,4.
Note that the point of intersection appears to be 3,4. We must now check the point 3,4 in both equations to see that it is a solution to the system. As a check we substitute the ordered pair 3,4 in each equation to see if we get a true statement. Are there any other points that would satisfy both equations? Not all pairs of equations will give a unique solution, as in this example. There are, in fact, three possibilities and you should be aware of them.
Since we are dealing with equations that graph as straight lines, we can examine these possibilities by observing graphs. Independent equations The two lines intersect in a single point.
In this case there is a unique solution. The example above was a system of independent equations. No matter how far these lines are extended, they will never intersect. Dependent equations The two equations give the same line. In this case any solution of one equation is a solution of the other.
In this case there will be infinitely many common solutions. In later algebra courses, methods of recognizing inconsistent and dependent equations will be learned. However, at this level we will deal only with independent equations.
You can then expect that all problems given in this chapter will have unique solutions. This means the graphs of all systems in this chapter will intersect in a single point. To solve a system of two linear equations by graphing 1. Make a table of values and sketch the graph of each equation on the same coordinate system. Find the values of x,y that name the point of intersection of the lines.
Check this point x,y in both equations. Again, in this table wc arbitrarily selected the values of x to be - 2, 0, and 5. Here we selected values for x to be 2, 4, and 6.
You could have chosen any values you wanted. We say "apparent" because we have not yet checked the ordered pair in both equations. Once it checks it is then definitely the solution. Graph two or more linear inequalities on the same set of coordinate axes. Determine the region of the plane that is the solution of the system. Later studies in mathematics will include the topic of linear programming.
Even though the topic itself is beyond the scope of this text, one technique used in linear programming is well within your reach-the graphing of systems of linear inequalities-and we will discuss it here. You found in the previous section that the solution to a system of linear equations is the intersection of the solutions to each of the equations. In the same manner the solution to a system of linear inequalities is the intersection of the half-planes and perhaps lines that are solutions to each individual linear inequality.
To graph the solution to this system we graph each linear inequality on the same set of coordinate axes and indicate the intersection of the two solution sets.
Note that the solution to a system of linear inequalities will be a collection of points. Again, use either a table of values or the slope-intercept form of the equation to graph the lines. The intersection of the two solution sets is that region of the plane in which the two screens intersect.
This region is shown in the graph. Note again that the solution does not include the lines. In section we solved a system of two equations with two unknowns by graphing. The graphical method is very useful, but it would not be practical if the solutions were fractions.
The actual point of intersection could be very difficult to determine. There are algebraic methods of solving systems.
In this section we will discuss the method of substitution. Example 1 Solve by the substitution method: Solution Step 1 We must solve for one unknown in one equation.
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