7.3 Solving Systems by Elimination

Transcription

1 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need a new propert for this new method. Equation Correspondence Propert If and then,. As we did in the last section, let s start with an eample. Eample : Solve First, notice that the equations have opposite coefficients for. Since this is the case, if we use the Equation Correspondence Propert to add the corresponding sides together, the s should cancel. This leaves us with a ver eas equation to solve. Dividing b on both sides will give us = Now we simpl need to get the -value that goes with it. So, just like in the previous section, we just have to plug this value into either original equation and solve. Let s use the first equation. + = + = - - = - Since the solution to a sstem is alwas written as an ordered pair, the solution here must be (, -) The process we used for solving the sstem in Eample is called the Elimination Method, since that is the primar goal, to eliminate one the variables. Here are the general steps for the Elimination Method. Solving a Sstem b Elimination Method. Write the equations in standard form, A + B = C.. If necessar, multipl one or both equations b a constant that will make the coefficients of one of the variables opposites of each other.. Add the corresponding sides of the equations and solve.. Substitute the value from step into either original equation and solve.. Write the solution as the ordered pair from steps and.. Check.

2 Now let s do some eamples. Eample : Solve b using the Elimination Method. a. b. c. a. First, we see that neither of the variables has opposite coefficients. This means we will have to multipl one (or possibl) both equations b a constant in order to get one of the variables to have opposite coefficients. So, we need to start b deciding which variable to eliminate. Either variable will work, but in this case, it seems like it might be easier to eliminate the s since the alread have opposite signs. This means we will have to multipl the top equation b, so that we can get a - which is opposite of the which is in the bottom equation. This gives Now we can add the sides as we did in Eample to eliminate the s. Then solve. Then substitute = into either original equation and solve for. Let s put it into the bottom equation. So the solution to the sstem is (0, ) b. Again, here we need to start b deciding which variable we want to eliminate. In this case, it looks like the s will be the best, since we would onl need to multipl the top equation b - to get the coefficients of to be -8 and 8. Now we add the sides, to eliminate the s and solve.

3 Even though the value is incredibl inconvenient, we can still get the that goes with it. We just plug it into one of the original equations and solve. Let s put it into the top equation. Clear the fractions b multipling b the LCD, 7 So the solution is. c. This time, the equations do not start in standard form. However, getting them to standard form is fairl eas. We simpl need to get all the variable terms to the same side. Now we have the sstem. So we see that neither of the variables is eas to eliminate. So we can just pick whichever we want to eliminate. Let s eliminate the s this time. To do so, we will have to multipl both equations b constants to get the coefficients to cancel. Clearl, we want the coefficients to be the LCM of and, that is. So we will multipl the top equation b and the bottom b. Now, we add the sides and solve. Then plug in = - to either original, standard form, equation, let s go with the bottom.

4 So the solution is (-, ). Just like alwas with sstems of linear equations, we can have one solution, no solution or infinite solutions. The equations in Eample all had one solution. So when using the Elimination Method what happens with the no solution and infinite solution cases. We will see in the net eample. Eample : Solve b using the Elimination Method. a. b. a. As we did in the previous eamples, we need to first decide which variable to eliminate. Clearl, the are both somewhat reasonable to eliminate. So, let s eliminate the s. To do so, we need to start b multipling the bottom equation b -. Adding corresponding sides gives us Just as we saw in the last section, this must mean the sstem has no solution. b. First we need to get the top equation into standard form. This is done b simpl subtracting from both sides. So we have Now we decide which variable to eliminate. The seem equall simple, so let s eliminate the s. So multipl the top equation b - Adding sides we have

5 So, as we saw in the last section, this must mean the sstem has infinite solutions. One of the biggest challenges in solving sstems of equations is determining which method to use. However, either method will work for ever sstem. That being said, sometimes one method is preferred over another. Usuall, if one of the variables is alread solved, the substitution method is best. If the equations are set up so that the are alread in standard form, however, then usuall elimination method is best. Graphing to solve a sstem is never a good idea. It is far to inaccurate b hand. Eample : Solve using an method. a. b. a. Since the equations are alread in standard form, the Elimination Method seems like the best choice to solve this sstem. So we start b eliminating the terms, b multipling the top b and the bottom b. Now add the sides and solve Plugging this into the top equation gives us So the solution is (-8, ). b. This time, it seems as though the Substitution Method would be best since the equations are both solved for. So since each left side equals, the right sides must be equal, which is the same as substituting the bottom equation into the top. We have

6 Now putting this value into the top equation will give us So the solution is (0, ). 7. Eercises Solve b using the Elimination Method

7 .... Solve b an method