CHAPTER 6: SYSTEMS OF TWO LINEAR EQUATIONS IN TWO ...

1 CHAPTER 6 171 CHAPTER 6: SYSTEMS OF TWO LINEAR EQUATIONS IN TWO VARIABLES Contents CHAPTER 6: SYSTEMS OF TWO LINEAR EQUATIONS IN TWO VARIABLES .. 171 SECTION : system OF EQUATIONS : GRAPHING .. 172 A. VERIFYING SOLUTIONS .. 172 B. SOLVE A system BY GRAPHING .. 173 EXERCISES .. 175 SECTION : SYSTEMS OF EQUATIONS : THE SUBSTITUTION METHOD .. 176 A. THE SUBSTITUTION 176 B. SUBSTITUTION: SPECIAL 178 EXERCISES .. 181 SECTION : system OF EQUATIONS : THE ADDITION METHOD .. 182 A. THE ADDITION METHOD .. 182 B. THE ADDITION METHOD WITH MULTIPLICATION .. 183 C. MULTIPLYING TWO 185 D. ADDITION: SPECIAL CASES .. 186 EXERCISES .. 189 SECTION : APPLICATIONS WITH SYSTEMS OF EQUATIONS .

2 190 A. VALUE & INTEREST PROBLEMS .. 190 B. MIXTURE PROBLEMS .. 192 C. UNIFORM MOTION WITH UNKNOWN RATES .. 193 EXERCISES .. 195 CHAPTER REVEW .. 196 CHAPTER 6 172 SECTION : system OF EQUATIONS : GRAPHING A. VERIFYING SOLUTIONS In CHAPTER 2 we solved single variable LINEAR EQUATIONS . For example, to solve for given the LINEAR equation +3 =4, we must isolate the variable . This is done by moving any term with an to the left of the equal sign and the rest of the terms to the right of the equal sign. Doing this gives us =1. What if we wanted to solve for two variables? For example, if we wanted to solve for and in the equation + =3 To solve for all we would need to do is isolate it.

3 Doing so gives us = +3 The problem here is that even when is by itself, we still don t know what is equal to. Having on the other side of the equal sign puts us in a tough spot since we have no idea what is. It turns out that in order to solve a LINEAR equation of two variables, we will need another equation. In other words, to solve for two variables and , we will need two EQUATIONS . When solving for more than one equation and one variable, we call the set of EQUATIONS a system of EQUATIONS . When dealing with a system of EQUATIONS , we are looking for the values that make both EQUATIONS true.

4 If only one equation is true, then we have the wrong answer and must try again. A system of two LINEAR EQUATIONS in two variables is of the form + = + = Where , , , , , and are coefficients and and are variables. Given an ordered pair ( , ), we can check to see if this is a solution to a system by plugging the ordered pair into both EQUATIONS and verifying that both are true. For example, to verify that the point (4,1) is a solution to the system 12 + =3 34 + = 2 we will plug the point (4,1) into the first equation. 12(4)+(1)=3 2 +1 =3 3 =3 Seeing that this equation is true, let s verify the next one. 34(4)+(1)= 2 3 +1 = 2 2 = 2 Since both EQUATIONS are true, we say the point (4,1) is a solution to the system .

5 MEDIA LESSON Verifying solutions (Duration 2:18 ) View the video lesson, take notes and complete the problems below To solve a system of EQUATIONS we want to find the value of _____ and the value of _____ that satisfies _____ EQUATIONS . The point of intersection is the point that lies on _____ lines CHAPTER 6 173 Example: a) Given the graph, identify the solution to the system of EQUATIONS . Verify the solution. YOU TRY a) Is the ordered-pair (2,1) the solution to the system 3 =5 + =3 B.

6 SOLVE A system BY GRAPHING One way to solve a system of LINEAR EQUATIONS is by graphing each LINEAR equation on the same -plane. When this is done, one of three cases will arise: Case 1: Two Intersecting Lines If the two lines intersect at a single point, then there is one solution for the system : the point of intersection. Case 2: Parallel Lines If the two lines are parallel to each other (not touching), then there is no solution for the system . Case 3: Same Lines If one line is on top of the other line (equivalent lines), then there are infinitely many solutions for the system . MEDIA LESSON Solve by graphing (Duration 3:44) View the video lesson, take notes and complete the problems below -7-5-3-11357-5-4-3-2-1012345 CHAPTER 6 174 When we talk about solving a system of EQUATIONS what we re looking for is the combination of and that simultaneously make both equations_____.

7 Another way of saying that is it s the point _____ that lies on _____ lines at the same time. Example: a) Solve the following system by graphing. 2 + =5 3 = 8 YOU TRY a) Solve the system by graphing. If possible, write the solution as an ordered pair. 6 3 = 92 +2 = 6 b) Solve the system by graphing. If possible, write the solution as an ordered pair. 32 + = 4 32 + =1 CHAPTER 6 175 EXERCISES Solve each system by graphing. When possible, write the solution as an ordered pair.

8 1) = 3 = 4 2) =13 +2 = 53 4 3) +3 = 9 5 +3 =3 4) 2 +3 = 6 2 + =2 5) 2 + = 2 +3 =9 6) 2 = 1 0= 2 3 7) +7 =4 3 +7 =0 8) = 54 2 = 14 +2 9) =2 +2 = 4 10) =12 +4 =12 +1 11) 6 + = 3 + =2 12) +2 =6 5 4 =16 13) 2 + =4 2= +12 14) 16= 4 2 = 4 4 15) 5 +1 = + = 3 CHAPTER 6 176 SECTION : SYSTEMS OF EQUATIONS : THE SUBSTITUTION METHOD A. THE SUBSTITUTION METHOD In the previous section we saw that one way to solve a system of LINEAR EQUATIONS is to graph each equation on the same -plane. If the graph is not accurate, then it can be difficult to see the solution.

9 In this section we will look at another method to solving a system of LINEAR EQUATIONS : the substitution method. Let us look at an example. 6 3 = 9 ( ) 2 +2 = 6 ( ) To the right of each equation is a label. This will make keeping track of our work much easier. The idea behind the substitution method is to solve one equation for one variable, then substitute this value into the second equation. Once this substitution is made, the second equation becomes a one variable equation. Let us walk through the example above. Step 1: Solving for a variable (any variable you want) Looking at the first equation ( ), let us solve for.

10 6 3 = 9 3 = 6 9 =2 +3 So =2 +3. We don t know what is since there is an in it, but that s okay. Step 2: Substitution Now let us take a look at the second equation ( ). Since we solved for in the first equation, we know what is equal to. We will use this value and plug it into the second equation (the variable we are plugging into the equation is in red) 2 +2 = 6 2 +2(2 +3)= 6 Step 3: Simplify and solve Notice how the second equation changes with the substitution we performed above! Our two variable equation 2 +2 = 6 just became the one variable equation 2 +2(2 +3)= 6. And we know how to solve one variable EQUATIONS .