Section 3.5 Systems of Equations

What is a system of equations? Two or more equations in the same variables

To solve a system Find all ordered pairs (x, y) that make both equations true

Methods to Solve 1.Graphing method 2.Substitution method 3.Linear combination/Elimination method

Solutions 1.One solution, ordered pair (intersection of lines) 2.No solution (lines are parallel) 3.Infinitely many solutions (when same exact line)

Calculator Directions 1.Enter equations in y= y 1 =1 st line and y 2 =2 nd line 2.Hit graph to see lines, change window if needed 3.Hit 2 nd Trace (Calculate) 4.Hit or scroll down to Intersect (#5) 5.Hit enter 3 times to obtain solution 6.Write solution as an ordered pair

Graphing Method Graph each line on the same coordinate plane. If lines intersect, there is only one solution: the intersection point. If lines are parallel, there is no solution. If lines coincide, there are infinitely many solutions.

Substitution Method Uses substitution of one equation into the other to solve for the other variable Goal: Isolate one variable (if not already given) Hint: Isolate the variable that will allow for easy algebra!

Linear combination Method Add the equations Goal: To combine the equations to eliminate a variable Hint: Create coefficients that are opposites for one of the variables

New Vocab Consistent Equations- A system of equations with at least one solution –Dependent Equations- A consistent system with infinitely many solutions (coinciding lines) Inconsistent Equations- A system of equations with no solution

Use intersect, are parallel, or coincide to make a true statement 1.If two lines have the same slope and different y-intercepts, then the lines ___________. 1.If one equation can be obtained from another equation by multiplying both sides by the same nonzero number, then the graphs ___________. are parallel coincide

Use intersect, are parallel, or coincide to make a true statement 3.If two lines have different slopes and the same x-intercept, then the lines __________. 3.If two lines have more than one point in common, then the lines ___________. 5. If the system of equations are dependent, then the lines ___________. intersect coincide