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Section 3.5 Systems of Equations

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

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

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

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

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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

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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.

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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!

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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

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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

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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

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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

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