SOLVING SYSTEMS OF EQUATIONS IN TWO VARIABLES SWBAT: Solve systems of equations graphically Solve systems of equations algebraically.

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SOLVING SYSTEMS OF EQUATIONS IN TWO VARIABLES SWBAT: Solve systems of equations graphically Solve systems of equations algebraically

Schema Activator I am thinking of two numbers whose sum is 25 and whose difference is 9. 1.Write a system of equations that models this situation, then solve the system. 2.What are the two numbers?

Systems of Equations Is a set of two or more equations. To “solve” a system of equations means to find values for the variables in the equations that makes all the equations true. Ways to solve: – Graphically – Algebraically Substitution Elimination

Graphically When solving a system there are 3 possible types of solutions 1 SOLUTIONINFINITE SOLUTIONSNO SOLUTION When 2 lines cross at one point where they cross is the solution When 2 lines are on top of one another then they are the same line and have an infinite number of solutions When 2 lines never cross then they have no solution

STEPS: Solving systems of equations graphically 1.Put your equations in y=mx+b form. 2.Graph the equations: 1.Graph the y-intercept 2.Use the slope to graph a second point. 3.If the graphs intersect the point of intersection is the solution. If the graphs are parallel there is no solution. If the graphs are the same line there is an infinite number of solutions.

Example 1: Solve the system 1.Solve for y 2.Graph the 1 st equation 3.Graph the 2 nd equation 4.Determine the solution No Solution!!

Example 2: Solve the system 1.Solve for y 2.Graph the 1 st equation 3.Graph the 2 nd equation 4.Determine the solution Infinite Solutions!!

Example 3: Solve the system 1.Solve for y 2.Graph the 1 st equation 3.Graph the 2 nd equation 4.Determine the solution (-2, -7)

Algebraically: Substitution

Solving by Substitution 1. Solve either of the equations for one variable in terms of the other. 2. Substitute the expression found in step 1 into the other equation. This will result in an equation with one variable. 3. Solve the equation with one variable. 4. Back-substitute the value from step 3 into one of the original equations. Simplify. Check.

Example 1 Keiocha and Jaime are competing for babysitting jobs in their neighborhood. Keiocha charges $15 plus $2.00 per hour while Jaime charges $10 plus $3.00 per hour. 1.Write an equation representing the total cost y of Keiocha’s services where x represents the number of hours. 2.Write an equation representing the total cost y of Jaime’s services where x represents the number of hours. 3.Solve the system of equations using substitution. 4.What does the solution represent in terms of this problem?

Example 2 Jose has $70 in the bank and Franchesca has $148 in the bank. Jose plans to save $5 each week and Franchesca plans to save $3 each week. 1.Write an equation representing Jose’s bank account balance (y) after x weeks. 2.Write an equation representing Franchesca’s bank account balance (y) after x weeks. 3.After how many weeks will their bank account balances be the same? 4.How much will each account have?

Algebraically: Elimination

Solving by Elimination/Combination This is another method that you can use to solve a system of equations. The goal of this method is to eliminate or combine one of the variables by adding the equations.

Steps 1.Check to see if you can eliminate one of your variables. (For example: If one of the equations has 3x then the other equation would have a - 3x) 2.Multiply one or both of the equations by a constant to obtain coefficients that differ in sign. 3.Add the equations. (Combine like terms!) 4.Substitute the value obtained into either equation to find the other variable.

Example 3 Ms. Rosenbaum is giving her Algebra II class a test worth 100 points containing 40 questions. There are 2-point and 4-point questions on the test. 1.Write two equations that model this situation. 2.Solve the system to determine how many of each type of question are on the test. 3.What percentage of the test is composed of 4- point questions?

Example 4 I go to the grocery store and buy 5 apples and 8 oranges and it cost $3.75. Then I go back to the store later that day and buy 2 apples and 4 oranges and it cost $1.60. How much does each apple and orange cost?