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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. Graphing method Substitution method Linear combination/Elimination method.
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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 • Graphing method • Substitution method • Linear combination/Elimination method
Solutions • One solution, ordered pair (intersection of lines) • No solution (lines are parallel) • Infinitely many solutions (when same exact line)
Calculator Directions • Enter equations in y= • y1=1st line and y2=2nd line • Hit graph to see lines, change window if needed • Hit 2nd Trace (Calculate) • Hit or scroll down to Intersect (#5) • Hit enter 3 times to obtain solution • 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 • If two lines have the same slope and different y-intercepts, then the lines ___________. • 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 • If two lines have different slopes and the same x-intercept, then the lines __________. • 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 coincide