3-2 Solving Systems Algebraically SWBAT: 1) Solve systems of linear equations by using substitution 2) Solve real world problems by using systems of linear equations

Solve the System by Graphing… They intersect at (2,4), and both have it as a solution.

Wait a Minute… Notice in the first equation y = x + 2 Substitute in x + 2 for y in the second equation. Solve for x.

We found x! How about y? Well y = x + 2 and y = 3x – 2 We know x = 2 Substitute in x to either equation and solve for y! y = x + 2 y = y = 4 y = 3x – 2 y = 3(2) – 2 y = 6 – 2 y = 4 So the Solution is (2,4)

Steps for Substitution 1.Pick one equation, solve for one variable. (Solve in terms of x or y) 2.Substitute that expression equal to the variable into the other equation. Solve for the opposite variable. 3.Sub the solutions into one of the original equations and find the other solution. 4.Write your solution as an ordered pair.

Ex 1: Solve using Substitution

Ex 2: Compare Values

Ex 3: Consistent and Dependent Systems Since the variables eliminated and the end result is a true statement…this system has infinite solutions. It is Consistent and Dependent.

Ex 3:Inconsistent Since the end result was not balanced, there would be no solutions. (Parallel Lines – no points of intersection) The system is Inconsistent…

Ex 4: Word Problems Mr. Falcicchio spent 40 minutes icing 24 cupcakes. It took him 1 min to ice a vanilla cupcake and 2 minutes to ice a chocolate cupcake. How many of each cupcake was made? Use a System to solve. 8 vanilla and 16 chocolate cupcakes

Ex 4b: Word Problems Mr. Frew coaches the Swim Team. He has 3 times as many boys as girls. He has 88 swimmers. How many Boys and Girls are there? So Mr. Frew has 66 boy and 22 girl swimmers

Solving Systems of Equations using Elimination You will be able to solve systems of equations using previous methods as well as using elimination to solve for a variable.

Elimination using Addition Sometimes adding two equations together will eliminate one variable. Using this step is called elimination. Once we eliminate one variable, we can solve for the remaining variable. We will then substitute for that variable into one of the equations in the system, in order to solve for the remaining variable In order to use elimination the equations must be set up in Standard Form. (x and y on same side)

Elimination with Addition (3,5) is the solution Notice how the y variables are opposites… Add the two Equations together.

More Practice Problems 1.x + y = -3 x – y = 1 2.3m – 2n = 13 m + 2n = 7

Example using Elimination with Same Signs Notice how the t variables are equivalent… Subtract the two expressions (4,-7) is the solution

Ex 2: Subtraction w/ Addition? Notice the b variables have the exact same coefficient. Multiply one whole equation by -1 to change signs!

Ex 2B: Elimination

What if the Variables Don’t Match? What would we do if our system of equations did not have two variables with the same coefficient? Ex: 3x + 4y = 6 5x + 2y = -4 Can elimination still be used in order to solve the system of equations?

Remember Multiplying by -1? We don’t always have to multiply equations by the same value. Notice how the y- coefficients are multiples of 2. Multiply the bottom equation by -2. What happens? Can we use elimination? Explain…

Ex 3: Solving Using Elimination

Determine the Best Method for Solving the System of Equations 9x – 8y = 424x – 2y = 14 4x + 8y = -16y = x 6x – y = 91/2x – 2/3y = 7/3 6x – y = 113/2 x + 2y = -25

Word Problem Find two numbers whose sum is 64 and whose difference is 42

Word Problem A youth group and their leaders visited Mammoth Cave. Two adults and 5 students in one van paid 77 dollars. Two adults and 7 students paid 95 dollars for the same tour. Find the adult and student prices.