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. Upon completing this section you should be able to: 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. We can choose either x or y in either the first or second equation.
Our choice can be based on obtaining the simplest expression. Look at both equations and see if either of them has a variable with a coefficient of one. Step 2 Substitute the value of x into the other equation.
Step 3 Solve for the unknown. Remember, first remove parentheses. Since we have already solved the second equation for x in terms of y, we may use it. Thus, we have the solution 2, Remember, x is written first in the ordered pair.
Step 5 Check the solution in both equations. Remember that the solution for a system must be true for each equation in the system.
Since the solution 2,-1 does check. Check this ordered pair in both equations. Neither of these equations had a variable with a coefficient of one. In this case, solving by substitution is not the best method, but we will do it that way just to show it can be done. The next section will give us an easier method. Upon completing this section you should be able to solve a system of two linear equations by the addition method. The addition method for solving a system of linear equations is based on two facts that we have used previously.
First we know that the solutions to an equation do not change if every term of that equation is multiplied by a nonzero number. Second we know that if we add the same or equal quantities to both sides of an equation, the results are still equal. Example 1 Solve by addition:. Note that we could solve this system by the substitution method, by solving the first equation for y. Solve this system by the substitution method and compare your solution with that obtained in this section.
Solution Step 1 Our purpose is to add the two equations and eliminate one of the unknowns so that we can solve the resulting equation in one unknown. If we add the equations as they are, we will not eliminate an unknown. This means we must first multiply each side of one or both of the equations by a number or numbers that will lead to the elimination of one of the unknowns when the equations are added.
After carefully looking at the problem, we note that the easiest unknown to eliminate is y. This is done by first multiplying each side of the first equation by Note that each term must be multiplied by - 2. Step 2 Add the equations. Step 3 Solve the resulting equation. In this case we simply multiply each side by Step 4 Find the value of the other unknown by substituting this value into one of the original equations.
Step 5 If we check the ordered pair 4,-3 in both equations, we see that it is a solution of the system. Example 2 Solve by addition:. Note that in this system no variable has a coefficient of one. Therefore, the best method of solving it is the addition method. Solution Step 1 Both equations will have to be changed to eliminate one of the unknowns. Neither unknown will be easier than the other, so choose to eliminate either x or y.
Rewrite the first two inequalities with y alone on one side. Consider a point that is not on the line - say, 0 , 0 - and substitute in the inequality. Shade upper half of the line. Here point 0 , 0 satisfies the inequality, so shade the half that contains the point.
The solution of the system of inequalities is the intersection region of the solutions of the three inequalities. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.
Graph showing the linear relationship between time and the change in revenue. All four—the table, the words, the equation, and the graph—are representations of the same thing. As you can see, graphs are especially good at conveying information about the relationship between things in this case, time and revenue with very little explanation. Graphs of Inequalities ». Graphs of Inequalities II ».
Math Skills Overview Guide. What does it look like? You'll use it
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