Solving Systems of Equations


 Trevor Craig
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1 Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that satisfies all of the equations in the sstem. Eample of a sstem: The ordered pair (, ) is a solution because: () () () 6 The ordered pair (, ) is a solution to the sstem because it is a solution for both equations.
2 The Substitution Method for Solving Sstems Solve the sstem of equations. Since = +, we can substitute + in for in the top equation without affecting its solution. We get one equation with a single variable. Solve the equation. ) ( Once is known, ou can backsubstitute = into the equation = + to find the value of. The solution is thus (, ). Check this in both of the original equations.
3 Check: The solution checks in both equations. Look at the graphs of the lines in the sstem (, ) **The solution to the sstem is the point where the graphs of the equations intersect.
4 Eample: Solve the sstem of equations. 9 We must begin b solving one of the equations for one of the variables. Solve the first equation for. (Alwas solve for the variable that has a coefficient of if possible.) 9 Substitute and solve the resulting equation. 9 9 ( ) 9 9 Backsubstitute to find. ( ) The solution is (, ).
5 The point of intersection of the lines is (, ). (,) Steps for the Substitution Method for Solving Sstems:. Solve one of the equations for one variable in terms of the other.. Substitute the epression found in Step into the other equation to obtain an equation in one variable.. Solve the equation obtained in Step.. Backsubstitute the value obtained in Step into the epression obtained in Step to find the value of the other variable.. Check that the solution satisfies each of the original equations.
6 6 Eample: Solve the sstem of equations. Solution: 0 ) )( ( 0 ) ( 0 ) ( or We must backsubstitute each of these values of to solve for the corresponding values of.
7 Backsubstitute: = =  ( ) The solutions are (, ) and (, ). Look at the graphs of these equations. (, ) (, ) The solution to the sstem is the set of points where the graphs of the equations intersect. 7
8 8 Eample: Solve the sstem of equations. Solution: 0 ) ( This does not factor, so use the quadratic formula. () ()() ) ( a ac b b This sstem has no real solutions.
9 Look at the graph of the sstem These graphs do not intersect. There is no real solution to this sstem, as we found b solving algebraicall. A sstem of equations in unknowns can have:. Eactl one real solution. More than one real solution. No real solution. 9
10 Solving Sstems Graphicall Eample: Solve the sstem using a graphing calculator. Solution:. Enter each equation in the [=] screen.. Press [GRAPH].. Press [ nd ] [CALC] and choose [intersect].. The cursor will appear on the graph of the first equation entered. Use the left and right arrows to move the cursor as close the visual point of intersection as possible. Press [ENTER].. The cursor will jump to the graph of the nd equation. Use the left and right arrows again to move the cursor to the visual point of intersection. Press [ENTER]. 6. The calculator screen reads Guess? Press [ENTER]. 7. The point of intersection is given. Repeat the process for an other visual points of intersection. The solution to the sstem is (0, ) and (, ). Note: Using the up and down arrows will move the cursor from one equation to the other. The right and left arrows move the cursor along one of the equations. 0
11 Eample: Solve the sstem of equations graphicall. ln CHAT PreCalculus Note: If we tried to solve this b substitution, we would get the equation + ln =, which is difficult to solve using standard algebraic techniques. (, 0) Check: ln 0 ln The solution (, 0) checks in both equations.
12 Application Eample: A compan has fied monthl manufacturing costs of $,000, and it costs $0.9 to produce each unit of product. The compan then sells each unit for $.. How man units must be sold before this compan breaks even? (The breakeven point is where the cost equals the revenue.) Solution: The total cost of producing units is: Total Cost Cost per Number =. unit of units + Fied cost For our problem, C = 0.9 +,000 The total revenue obtained b selling units is: Total Revenue Price per =. unit Number of units For our problem, R =.
13 Our sstem of equations looks like: C R. Because the breakeven point occurs when R = C, we are reall looking at the sstem: C C. Solve this sstem b substitution , ,000 The breakeven point occurs when 0,000 units are produced.
14 Question: How man units must be sold to make a profit of $7,000? Profit is found b CHAT PreCalculus Profit = Revenue  Cost Revenue = the mone received from selling the product Cost = the mone spent to produce the product For our problem C R P P P R C. (0.9,000) 0.,000 We want the profit to be $7,000 so substitute that for P. P 0.,000 7,000 0.,000 9, ,000 0,000 units must be produced for a profit of $7,000.
15 Additional Eamples: Eample: Solve the sstem. Solution: (, ) Eample: Solve the sstem. Solution: (0, ), (, ), (, 0) Eample: Solve the sstem. Solution: (, )
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