13.1 2X2 Systems of Equations


 Regina Logan
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1 . X Sstems of Equations In this section we want to spend some time reviewing sstems of equations. Recall there are two basic techniques we use for solving a sstem of equations: Elimination and Substitution. We will start with the elimination method. The ke idea in the elimination method is to eliminate one of the two variables b making the coefficients opposites on the two equations and adding the equations together. We will illustrate this method in the following eample. Eample : Solve using the elimination method. a. 9 b. 9 a. So to use the elimination method to solve, we need to start b determining which variable we would like eliminate. Lets choose the variable since the signs are alread opposite. Net we will need to multipl the top equation b a to make the coefficients opposites. Then we simpl add equations and solve. We get 9 9 So now we need the value of the solution. So we can simpl substitute the value into one of the original equations and solve for. Let s use the first equation. We get Since the solution to a sstem is an ordered pair we write the solution is,. b. This time we need to start b putting the equations in standard form. Then we will proceed as above. 9 9 We will choose to eliminate the s 9 Clear fractions b multipling b
2 Now we solve for. We use the second equation this time. We get So the solution is,. Net we will review the substitution method. The ke idea in the substitution method is to solve one of the equations for one of the variables and then substitute that epression into the other equation. We illustrate this in the following eample. Eample : Solve using the substitution method. a. b. a. To use the substitution method we need to solve one equation for one variable. In this case, the bottom equation is alread solved for. Net, we substitute that epression into the other equation and solve for the remaining variable. We proceed as follows Now we simpl substitute this value back into the original equation for, and solve. We get So our solution is,. b. So we need to start b solving one of the equation for one of the variables. We generall choose the easiest variable to get to. In this case that is the on the bottom equation. So we solve for it. We get
3 Now substitute as before and solve. Again, substitute this value into the epression we obtained for. We get So our solution is,. Now, recall that graphicall, the solution to a sstem of equations is the points of intersection of the graphs of the equations. So since, in this section, we are dealing onl with lines, there are onl three was that the can intersect, that is, there are onl three different possibilities for tpes of solutions to these sstems. One intersection point No intersection point Ever point intersects = One solution = No solution = Infinite solutions In order to determine if we have one of the two last cases we merel need to tr to solve and interpret what we find. Eample : Solve the following using an method. a. b a. First we will clear the fractions to evaluate which method of solving is best. So the first equation will be
4 Now, lets us the elimination method to solve. We will eliminate the s. Since all the variables eliminated we must have a special case. Since = for all values of and, we have the infinite solutions case. We simpl sa that the sstem has infinite solutions. b. Again lets use the elimination method for solving. Again we will eliminate the. We multipl the top equation b. and the bottom equation b.. We get Once again we have eliminated all the variables. In this case however we are left with a statement that is not true. Since that is the case we have no solution. Of course there are man ver useful applications of sstems of equations. We will concentrate on just a few different tpes: the miture problem, the investment problem, the moving object problem. We have reviewed the basic moving object problem in section R.. The difference here is that we can now use two different variables to solve. Eample : A motorboat traveling with the current went km in h. Against the current, the boat could go onl km in the same amount of time. Find the rate of the boat in calm water and the rate of the current. First we need to set up our variables. Lets let r be the rate of the boat in calm water and c be the rate of the current. That means that the total rate of the boat with the current would be r + c and the total rate against the current would be r c. So, like we did in section R. we will set up a table to summarize our data and construct a sstem of equations. With current Against current Now using the formula we get the sstem r t d r + c r  c r c r c. We need onl solve the sstem. Lets use elimination. We will start b divide the each equation b on both sides to simplif the sstem. r c r c
5 r c r c r r So the rate of the boat is km/hour. To find the rate of the current, c, we simpl substitute it back into one of the equations and solve. We get So the current is km/hour. c c For the investment problem we simpl use the formula I P r t. Since our time will ver often be one ear, we can reall just use I P r. We use a table just like we will with the miture problems. There are two major tpes of miture problems: Total Value and Total Quantit. The wa we tell the difference is b the rate we are given. If the problem gives us a monetar rate (like $ per pound) then we have a total value problem. If we have a percentage rate then we have a total quantit problem. We use the following formulas in conjunction with a table to solve: Amount Rate = Total Value (or simpl A r TV ) or Amount Rate = Quantit (or simpl A r Q ). We will illustrate each tpe. Eample : At the theatre, Jon bus boes of popcorn and soft drinks for $.. One bo of popcorn and one soft drink would cost $.. What is the cost of a bo of popcorn? Since we are dealing with mone here we use the formula A r TV. Again, we will make a table for our data. We will let the cost of one bo of popcorn be p and cost for one soft drink be d. We fill in the table with two different situations, the purchase that Jon makes and a purchase that would be onl one of each. We get the following. A r TV Jon s purchase popcorn soft drinks p d p d One of each popcorn soft drinks p d p d The bo in the top right is represents the total value of Jon s purchase and the bo in the bottom right is the total value of a purchase of one or each. Therefore we get the sstem p d. p d. Lets solve this b substitution. Solving the bottom for p gives. d d.. d d. d. p. d. Substituting we get So soft drinks are $. each. Now we can solve for cost of popcorn. Substituting we get p... Thus popcorn costs $. each.
6 Eample : A chemist wishes to make liters of a.% acid solution b miing a % solution with a.% solution. How man liters of each solution are necessar? This time we will need to use the formula A r Q since we are given percentage rates. Also, we will need to use three rows, one for the.% solution, one for the % solution and one for the final solution. Lets call the amount of % solution and the amount of.% solution. So we fill in the table as follows. % solution.% solution Final solution A r..... ()(.) The wa that this tpe of miture problem works is the final amount must alwas equal the sum of the amounts of the two being mied, and the final quantit must alwas equal the sum of the two quantities. So that being said we get the sstem.. We will solve this sstem b the elimination method..... Q Therefore,,. 9. So we need about,.9 liters of % solution and. liters of.% solution.. Eercises Solve the following using the elimination method
7 Solve the following using the substitution method Solve using an method Set up a sstem for the following problems and solve using an method.. A boat travels miles in hours upstream. In the same amount of time the boat can travel miles downstream. Find the rate of the current and the rate of the boat in still water.. A foot rope is cut in two pieces so that one piece is twice as long as the other. How long is each piece?. How man grams of silver that is % pure must be mied together with silver that is % pure to obtain a miture of 9 grams of silver that is % pure?
8 . At a barbecue, there were dinners served. Children s plates were $. each and adult s plates were $. each. If the total amount of mone collected for dinners at the barbecue was $, how man of each tpe of plate was served? 9. A store sells shirts, one kind at $. and the other at $9.. In all, $9. was taken in. How man of each tpe were sold?. A store buer purchased regular calculators and graphing calculators for a total cost of $.. A second purchase, at the same prices, included regular calculators and graphing calculators for a total of $.. Find the cost for each tpe of calculator.. Fling with the wind, a plane flew miles in hours. Fling against the wind, the plane could fl onl miles in the same amount of time. Find the rate of the plane in calm air and the rate of the wind.. A plane traveling with the wind flew miles in. hours. Against the wind, the plane required. hours to go the same distance. Find the rate of the plane in calm air and the rate of the wind.. A rowing team rowing with the current traveled mi in h. Against the current, the team rowed mi in h. Find the rate of the rowing team in calm water and the rate of the current.. How man grams of pure acid must be added to % acid to make 9 grams of solution that is % acid?. A chemist has some % hdrogen peroide solution and some % hdrogen peroide solution. How man milliliters of the % solution should be used to make milliliters of solution that is.% hdrogen peroide?. How man grams of % pure gold must be mied with % pure gold to make grams of % pure gold?. A total of $ is deposited into two simple interest accounts. One account has an interest rate of.% and the other has a rate of.%. How much should be invested in the.% account to earn a total interest of $?. A total of $, is invested to provide retirement income. Part of the $, is invested in an account paing 9% interest. How much should be invested into an % interest account so that the total income is $? 9. A total of $, is invested into two simple interest accounts. One account has an interest rate of % while the other account has a rate of %. How much must be invested in each account so that both accounts earn the same amount of interest?. Find the cost per kilogram of a grated cheese miture made from kg of cheese that costs $. per kilogram and kg of cheese that costs $. per kilogram.. How man grams of pure acid must be added to g of a % acid solution to make a solution that is % acid?. A researcher mies g of % aluminum allo with g of a % aluminum allo. What is the percent concentration of the resulting allo?
9 . A total of $9 is deposited into two simple interest accounts. On one account the annual simple interest rate is %; on the second account the annual simple interest rate is %. How much should be invested in the % account so that the total interest earned is $?. An investment of $ is made into a % simple interest account. How much additional mone must be deposited into a.% simple interest account so that the total interest earned on both accounts is % of the total investment?. A jet plane traveling at mph overtakes a propellerdriven plane that has a. hour head start. The propellerdriven plane is traveling at a rate of mph. How far from the starting point does the jet overtake the propellerdriven plane?. A student pilot flies to a cit at an average speed of mph and then returns at an average speed of mph. Find the total distance between the two cities if the total fling time was hours.
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