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 1.Solve both equations for y 2.Graph both of the lines a.Plot y-int b.Do slope from y-int c.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 Solutions1 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

Substitution 1.Solve one equation for a variable, look for a variable with a 1 in front of it or an even number 2.Substitute the expression you just solved for into the variable of the other equation 3.Solve for the remaining variable 4.Take value of variable and substitute it into the first equation 5.Check ordered pair in second equation 6.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 1.Line up equations so variables are above one another 2.Determine what to multiply the equations by so when you add them one of the variables will cancel 3.Add the equations together 4.Solve for remaining variable 5.Sub in value of variable into original 1 st equation and solve 6.Check answer in second equation 7.Write answer as an ordered pair

Example