Essential Understanding andObjectives• Essential Understanding: you can solve a system of equations by writing equivalent systems until the value of one variable is clear. Then substitute to find the values of the other variable• Objectives:• Students will be able to solve linear systems algebraically
Iowa Core Curriculum• Algebra• A. CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.• A.CED.3 . Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.• A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.• A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Substitution Method• Use this method to solve a system of equations when it is easy to isolate one of the variables.• After isolating one of the variables, substitute for that variable in the other equation.• Then solve for the other variable
Example• What is the solution to the system of equations? ì3x + 4y = 12 í î2x + y = 10• Step 1: solve the equation for one of the variables• Step 2: Substitute the expression for y in the other equation. Then solve for x• Step 3: Substitute the value for x into one of the original equations. Solve for y.
Example• What is the solution of the system of equations? ì x + 3y = 5 í î-2x - 4y = -5
Example• An online music company offers 15 downloads for $19.75 and 40 downloads for $43.50. Each price includes the same one-time registration fee. What is the cost of each download and the registration fee?• Step 1: Relate • 15 (cost of one download) + one-time fee = $19.75 • 40 (cost of one download) + one-time fee = $43.50• Step 2: Define • Let c = cost of one download • Let f = the one time fee• Step 3: Write • 15c+ f = 19.75 • 40c + f= 43.50• Step 4: Solve using substitution• Solution: $.95 per download and $5.50 for the one time fee
Elimination Method• You can use the Addition Property of Equality (If a = b then a + c = b + c )to solve a system of equations. By adding a pair of additive inverses or subtract identical terms you can eliminate a variable.
Example• What is the solution of the system of equations? ì 4x + 2y = 9 í î-4x + 3y = 16• Step 1: Do you need to multiply to get equivalent terms?• Step 2: Add the equations together• Step 3: Solve for the remaining variable• Step 4: Chose one of the original equations and substitute the variable you already solved for.• Step 5: Solve for the other variable
Example• What is the solution of the system of equations? ì-2x + 8y = -8 í î5x - 8y = 20
Equivalent Systems• When you multiply each side of one or both equations in the same system by the same nonzero number, the new system and the original system has the same solutions. This is known as Equivalent Systems• Example 4• What is the solution of the system of equations? ì2x + 7y = 4 í î3x + 5y = -5
What are the solutions of thefollowing systems? Explain.ì5x + 2y = -4 ì-x + y = -2í íî3x + 7y = 15 î2x - 2y = 0 ì 4x + y = 6 í î12x + 3y = 18