2 equations 2 Variables Graphically Substitution Elimination Systems of equations 2 equations 2 Variables Graphically Substitution Elimination

Solving a System Getting the ordered pair that makes both equations true Where the two lines would intersect You could have: No solution – meaning they never intersect 1 solution they intersect once Infinite solutions – they are really the same line

Graphically Solve both equations for y Graph both of the lines Plot y-int Do slope from y-int Connect with a line 3. Find point of intersection This works well if lines intersect at an exact point

Example Solve the following system graphically Both already solved for y so graph them They intersect at (1,-2) Answer should be an Ordered pair

Number of solutions What would graph look like for No Solutions 1 Solution Infinite solutions Equations?

Graphically This is used a lot in word problems Determine which company is better to go with at a certain point and why, intersections tell you this.

Example Your niece wants to have a birthday party on either McDonalds or Dave and Busters. Given the information below set up a system of equations and graph them McDonalds: The fixed cost to set up is $75 and $5 for each person attending. Write an equation in slope intercept form to represent this. Dave and Busters: The fixed cost to set up is $50 and $10 for each person attending. Write an equation in slope intercept form to represent this. Once you have them graphed answer the following questions. At what number of guest is the cost equal? How do you know? What is the difference in cost for 20 people to go to Dave and Busters vs McDonalds? When is it better to go to Dave and Busters vs McDonalds and when is it better to go to McDonalds vs Dave and Busters? You want to invite 24 people to the part. How much will it cost at each location? You have a $250 limit on the cost of the party, how many people could you invite to each location?

Substitution Solve one equation for a variable, look for a variable with a 1 in front of it or an even number Substitute the expression you just solved for into the variable of the other equation Solve for the remaining variable Take value of variable and substitute it into the first equation Check ordered pair in second equation Write answer as an ordered pair

Example

Number of solutions How do you determine how many solutions if not graphing No Solutions – when you solve you get 2 numbers that are not equal to each other 1 Solution x and y both equal in number, process works Infinite solutions – x=x, number equals itself

Solutions Think back to the equations used for solving graphically If we solve them using substitution

Elimination Add the equations together to cancel out one of the variables Line up equations so variables are above one another Determine what to multiply the equations by so when you add them one of the variables will cancel Add the equations together Solve for remaining variable Sub in value of variable into original 1st equation and solve Check answer in second equation Write answer as an ordered pair

Example