Solving a System of 2 Linear Equations

Solving the SystemMethod 1: MatricesUsing the system of equations:5c + 4s = 60 c + 2s = 18Re-write as a matrix equation:

DefinitionA system of 2 linear equations is: 2 linear equations, both of which contain the same 2 variables.

Example: 5x + 4y = 60 (Equation #1) x + 2y = 18(Equation #2)Real-World ExampleMr. K goes to the concession stand at a baseball game. He buys 5 cheesesteaks and 4 large sodas and pays a total of 60 dollars. Later in the game, Mr. S goes to the concession stand. Mr. S buys 1 cheesesteak and 2 large sodas and pays a total of 18 dollars. What is the cost of one cheesesteak, and the cost of one large soda?SolutionTo find a solution to a system of 2 linear equations means to find the point(s) in common (if possible) between the two equations.In our example, this means to find the cost of one cheesesteak and the cost of one large soda.

NOTE: Sometimes a solution does not exist, or there are an infinite number of solutions. Setting Up the SystemTo set-up a system, do the following:Identify the variables (unknowns) in the problem.Example: Let c = Cost of one Cheesesteak Let s = Cost of one large SodaWrite 2 separate equations, using the variables above, to model the situation. Example: Each trip to the concession stand is an equation:Mr. Ks trip:5c + 4s = 60Mr Ss trip: c + 2s = 18Solving the System This tutorial will use 6 different methods to solve a system of 2 linear equations:MatricesGraphing InterceptsGraphing Functions (y =)Linear CombinationSubstitution One Equation into the OtherSubstitution Equations Equal to Each Other

Solving the SystemMethod 1: Matrices (Continued)Solve the system using inverse matrices:

Therefore, cost of one cheesesteak is c = $8,and the cost of one large soda is s = $5.

Solving the SystemMethod 2: Graphing - InterceptsFor each equation:Find the x-intercept: Point (x,0)Find the y-intercept: Point (0,y)

Let x = the cost of a cheesesteakLet y = the cost of a large sodaThen our system becomes

5x + 4y = 60 x + 2y = 18

Solving the SystemMethod 2: Graphing Intercepts5x + 4y = 60 (Equation 1)

X-intercept: 5x + 4(0) = 60 5x = 60 5 5 x = 12 Pt. (12,0)

Y-intercept:5(0) + 4y = 60 4y =60 4 4y = 15 Pt. (0,15)

x + 2y = 18 (Equation 2)

X-intercept:x + 2(0) = 18x =18 Pt. (18,0)

Y-intercept:(0) + 2y = 182y = 18 2 2 y = 9 Pt. (0, 9)5x + 4y = 60 (Equation 1) x + 2y = 18 (Equation 2)Solving the SystemMethod 2: Graphing Intercepts (Cont.)Step 1: Graph Equation 1 Using the intercepts

Solving the SystemMethod 2: Graphing Intercepts (Cont.)Step 2: Graph Equation 2 Using the intercepts

Solving the SystemMethod 2: Graphing Intercepts (Cont.)Step 2: Graph Equation 2 Using the intercepts

Solving the SystemMethod 2: Graphing Intercepts (Cont.)Step 3: Find the intersection - the solution.

Solving the SystemMethod 2: Graphing Intercepts (Cont.)Step 3: Find the intersection - the solution.

Solving the SystemMethod 2: Graphing Intercepts (Cont.)The intersection point (8,5) is the solution to the system of equations.

8, the x-coordinate, is the cost of a cheesesteak = $85, the y-coordinate, is the cost of a large soda = $5Solving the SystemMethod 3: Graphing - Functions (y =)Let x = the cost of a cheesesteakLet y = the cost of a large sodaThen our system becomes

5x + 4y = 60 x + 2y = 18

Now, we must solve each equation for ySolving the SystemMethod 3: Graphing - Functions (y =)5x + 4y = 60 (Equation 1)-5x -5x 4y = 60 5x 4 4

x + 2y = 18 (Equation 2)-x -x 2y = 18 - x 2 2

Solving the SystemMethod 3: Graphing - Functions (y =)Type each equation into y = in calculator

Set an appropriate WINDOWFor example: Xmin = -5, Xmax = 20, Xscl = 5Ymin = -5, Ymax = 20, Yscl = 5, Xres = 1Press GRAPH

Find the intersection point

Solving the SystemMethod 3: Graphing - Functions (y =)Using the Window above, graph should look like:

Solving the SystemMethod 3: Graphing - Functions (y =)To find the intersection point:Press 2nd TRACE 5:intersect ENTERPress ENTER 3 more timesSolution is:Intersection:X = 8Y = 5The ordered pair (8,5)8, the x-coordinate, is the cost of a cheesesteak = $85, the y-coordinate, is the cost of a large soda = $5

Solving the SystemMethod 4: Linear CombinationIn our equations: we can cancel the +4s in equation 1, by multiplying equation 2 by (-2).5c + 4s = 605c + 4s = 60-2 (c + 2s = 18) -2c +-4s = -36 (Add Equations) 3c = 24(Solve for c) 3c = 24 3 3 c = 8 (Cost of a cheesesteak)

Now use this to solve for the other variable.

Solving the SystemMethod 4: Linear CombinationYou can substitute c = 8 into either of the original equations, and solve for the other variable.Equation 1Equation 25c + 4s = 60c + 2s = 185(8) + 4s = 608 + 2s = 1840 + 4s = 60-8 -8-40 -40 2s = 10 4s = 20 2 2 4 4 s = 5s = 5So, s, the cost of a soda = $5. And, the solution of: c = 8 and s = 5, satisfies both original equations.

Solving the SystemMethod 5: Substitution One Equation into the OtherSolving equation 2 for c:c + 2s = 18 - 2s - 2sc = 18 2s (NOTE: This is NOT 16s)

Now, substitute this new expression for c into equation 1 for c:5c + 4s = 605(18 2s) + 4s = 60

Solving the SystemMethod 5: Substitution One Equation into the OtherNow, solve this equation for s:5(18 2s) + 4s = 6090 10s + 4s = 6090 6s = 60 -90 - 90 - 6s = - 30-6-6 s = 5

Solving the SystemMethod 5: Substitution One Equation into the OtherNow, substitute s = 5 into either of the original equations and solve for c.5c + 4s = 60(equation 1)5c + 4(5) = 605c + 20 = 60 - 20 - 205c = 405 5c = 8So, s, the cost of a soda = $5. And, c, the cost of a cheesesteak = $8.

Solving the SystemMethod 6: Substitution Equations Equal to Each Other5c + 4s = 60 (equation 1)c + 2s = 18 (equation 2)Solving for c:5c + 4s = 60 - 4s - 4s5c = 60 4s5 5 c = 60 - 4s 5 5 OR c = 12 0.8s

Solving for c:c + 2s = 18 - 2s - 2sc = 18 2s

Solving the SystemMethod 6: Substitution Equations Equal to Each OtherNow, set the two expression for c equal to each other and solve for s:12 0.8s = 18 2s + 0.8s + 0.8s12 = 18 1.2s-18 -18-6 = -1.2s-1.2 -1.2 5 = sSolving the SystemMethod 6: Substitution Equations Equal to Each OtherNow substitute s = 5 into either of the expressions found previously for c.c = 18 2sc = 18 2(5)c = 18 10c = 8

So, s, the cost of a soda = $5. And, c, the cost of a cheesesteak = $8.

Verifying the Solution to the SystemThe Check-StepAll of the methods of solving the system resulted in the same solution: s = 5, c = 8.To verify this solution satisfies both original equations, substitute the solution values into the original equations.Equation 1:Equation 2:5c + 4s = 60c + 2s = 185(8) + 4(5) = 608 + 2(5) = 1840 + 20 = 608 + 10 = 18 60 = 60 18 = 18

IMPORTANTIn the example given in this tutorial, one solution exists. However, not all systems of 2 linear equations have exactly one solution. It is possible to have either:An infinite number of solutions, ORNo solutionIf a system has an infinite number of solutions, this means that the equations are actually the same equation (the same line if graphing), just written in different forms.If a system has no solutions, this means that the equations have the same slopes, but different y-intercepts (parallel lines if graphing).

IMPORTANTTo determine if one solution exists, write each equation in Standard Form: Ax + By = C.For a system of 2 linear equations, this would look like: Ax + By = C (equation 1) Dx + Ey = F (equation 2)If the system has infinite solutions, then:

If the system has no solution, then:

Otherwise, the system has ONE solution.

Practice ProblemsSolve each of the following systems of linear equations by the method indicated. First, make sure the system does, in fact, have one solution. When finished with each problem, be sure to verify the solution found.

1.) Method: Matrices2x + 3y = 23x 4y = -14

Practice Problems2.) Method: Graphing - Intercepts 2x + 9y = 362x y = 16

3.) Method: Graphing Functions (y=)5x 6y = 482x + 5y = -3

Practice Problems4.) Method: Linear Combination4x 3y = 175x + 4y = 60

5.) Method: Substitution One Equation into the Other8x 9y = 194x + y = -7

Practice Problems6.) Method: Substitution Equations Equal to Each Other4x y = 63x + 2y = 21

7.) Method: You Choose8x 4y = 234x 2y = -17

Practice Problems8.) Method: You Choose-2x + 5y = 9y = 13 - x

9.) Method: You Choose5x + 3y = 12 15x + 9y = 36

Practice Problems10.) Method: You ChooseSuppose that the promotions manager of a minor league baseball team decided to have a giveaway of tote bags and t-shirts to the first 150 fans present. The team owner agrees to a budget of $1350 for the products to be given away. One bag costs $10 and one t-shirt costs $7. How many bags and how many t-shirts should be given away?

Set up the system of 2 linear equations and solve.