Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of Linear Equations by the Addition (Elimination) Method Application Problem Involving Two Equations and Two Variables. A system consists of more than one equation. There is more than one way to solve a system of equations. They are all valid methods, but sometimes one or another will be the better choice for solving a system of equations. The solutions of a system of equations are the values that make all the equations in the system true. When we check our answers, we must be sure to check them in all the equations, since a correct answer for one equation may not be correct for the others. The Graphing Method This will involve graphing the equations on a coordinate plane. The solution(s) to the system will be all points that are common to both graphs (where the graphs intersect). A system of two linear equations in two variables is two equations considered together. To solve a system is to find all the ordered pairs that satisfy both equations. When solving a system three situations can occur: 1. The lines intersect. There is one solution, the point where they intersect. 2. The lines are parallel. There is no solution, the lines do not intersect. 3. The lines are the same. There are infinitely many solutions, the lines coincide. Systems of equations can be solved by various methods which will be the focus of the unit. Example 1. a) Use the graphing approach to solve the system. Tell whether the system has one, infinite or no solutions. 2x - y = 5 3x + 2y = 4 Graph the two lines using any method. 2x y = 5 3x + 2y = 4 y x y x y x The solution to the system is the point of intersection: (, ).
Example 2: 2x + 3y = 5 x 5y = -17 The first step will be to graph these two equations. 2x + 3y = 5 It looks like the answer is (-2, 3) x 5y = -17 After we graph the lines on the same coordinate plane, we notice that they intersect in a point. It looks like (-2, 3) is the solution to this system, but we need to check this potential solution. In equation #1: 2(-2) + 3(3) = -4 + 9 = 5 Checks In equation #2: (-2) 5(3) = -2 15 = -17 Checks (-2, 3) Answer Note: The graphing method is not a very accurate one. If our graphs are not perfect (or the answers not integers), we will probably not be able to get the right answer. Graphing is useful, however, for visualizing the problem, estimating answers, and sometimes for informing us of how many answers we should expect to find. Note also: If the two linear graphs do not intersect, we call the system inconsistent (this means the lines are parallel). If the two linear graphs turn out to be the same line (there are infinitely many solutions: all the points on the line), then the system is consistent and dependent. If the two linear graphs intersect in a point, the system is consistent and independent.
The Substitution Method To solve a system of equations by the substitution method: 1. Solve one of the equations for one of the variables.(choose a variable with a coefficient of 1 or -1 if possible.) 2. Substitute this expression into the other equation to produce an equation with only one variable. 3. Solve the equation in Step 2 for the remaining variable. 4. Substitute this solution into the expression obtained in Step 1. 5. Solve for the second variable. 6. Write your solution set as an ordered pair, and check in each equation. Example 2x - 3y = 5 Solve the system by the substitution method: 3x + y = 2 y = 2-3x 1) Solve the 2nd equation for y. 2x - 3( ) = 5 2) Substitute 2-3x into the other equation in place of y. 2x - + = 5 3) Solve for x. 11x = x = y = 2-3( ) 4) Substitute this value into step 1. y = 5) Solve for y. {(, )} 6) Write the solution set. Check: 2( ) - 3( ) = 5? 7) Check the solution in both equations. 3( ) + ( ) = 2?
Elimination-by-Addition Method In the elimination-by-addition method we use the following operations to produce an equivalent system which can be easily solved. 1. Any two equations can be interchanged. 2. Both sides of an equation can be multiplied by any nonzero real number. 3. Any equation can be replaced by the sum of that equation and a nonzero multiple of another equation. Steps for elimination-by-addition method: 1. Write each equation in standard form if needed. 2. If necessary, multiply one or both equations by some constant which will make the x or y coefficients opposites. 3. Add the equations from step 2 together eliminating one of the variables. 4. Solve for the remaining variable. 5. Substitute this solution into either of the original equations. 6. Solve for the second variable. 7. Write the solution set and check. Example Solve the system by the elimination-by-addition method: x + 3y =7 2x - 3y = 5 The equations are in standard form. 3x = x = Solve for x. + 3y = 7 Substitute x = 4 into either equation. 3y = Solve for y. y = When we add the 2 equations the y's will cancel. {(, )} Write the solution set and check. check: 4 + 3(1) = 7? 2(4) - 3(1) = 5?
No solution and infinite solutions No Solution: Sometimes you will not have a solution (an ordered pair) to a system of equations this occurs when the lines represented by the equations never intersect. What this looks like when you solve the system is a result that is. For example your solution might be 12 = 14 which is false! In other words, the solution set is Ǿ and the system has no solution. Infinite Solutions: Sometimes your solution shows that one value is equal to the same value. For example, your solution might be 12 = 12. In this case, the solution set to the system is infinite. In other words, the equation is always true and any real number will solve the system. This occurs in linear equations when both equations represent the same line. In this case, the solution set is all real numbers. Which Method to Use? 1. If one equation is already solved for one of the variables, substitution would probably be the easiest method. 2. If solving for either variable in either equation would produce fractions to substitute, use the addition method. 3. Always clear fractions in both equations before deciding which method to use. Practice Problems - Solve by either substitution or elimination method: 5x - y = 13 1. 3x + 2y = 0 4x - 3y = 19 2. 5x + 4y = -15
y = 2x +7 3. 4x - 2y = 12 2x + y = 9 4. 4x + 2y = 18 3x + y = 0 5. x - 2y = -7
Systems that do not have solutions Sometimes a linear system will not have any solutions. Here s a system that has no solutions: x + y = 4 2x + 2y = 10 What happens in our system solving methods, if the system actually has no solution? Substitution Method: Here s what happens for the example problem. First equation solved for y: y = 4 x Substitute into second equation: 2x + 2(4 x) = 10 Distribute and simplify: 2x + 8 2x = 10 8 = 10 We end up with an always-false equation, which tells us that there s no solution. Elimination Method: Here s what happens for the example problem. First equation multiplied by 2: 2x + 2y = 8 Second equation: 2x + 2y = 10 Subtract: 0x + 0y = 2 0 = 2 We end up with an always-false equation, which tells us that there s no solution. In general, using either of the methods, the indicator that a system has no solutions is that you reach an equation that is always false, such as a number equaling a different number. Since the equation is false no matter what values x and y have, that means the system has no solution. You try it 15. a. Solve by substitution: b. Solve by elimination: 2x + 4y = 20 2x + 4y = 20 3x + 6y = 50 3x + 6y = 50
Systems that have infinitely many solutions Sometimes a linear system will have infinitely many solutions. Here s such a system: x + y = 4 2x + 2y = 8 What happens in our system solving methods, if the system has infinitely many solutions? Substitution Method: Here s what happens for the example problem. First equation solved for y: y = 4 x Substitute into second equation: 2x + 2(4 x) = 8 Distribute and simplify: 2x + 8 2x = 8 8 = 8 We get an always-true equation, which tells that there are infinitely many solutions. Elimination Method: Here s what happens for the example problem. First equation multiplied by 2: 2x + 2y = 8 Second equation: 2x + 2y = 8 Subtract: 0x + 0y = 0 0 = 0 We get an always-true equation, which tells that there are infinitely many solutions. In general, using either of the methods, the indicator that a system has infinitely many solutions is that you reach an equation that is always true, such as a number equaling the same number. When that happens, any of the infinitely many (x, y) pairs that makes one of the equations true also makes the other equation true.