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Section 3.5 Systems of Equations

Learn how to solve systems of equations using graphing, substitution, and linear combination methods. Understand different solutions, such as one solution, no solution, or infinitely many solutions. Practice with a calculator to find ordered pairs.

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Section 3.5 Systems of Equations

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  1. Section 3.5Systems of Equations

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

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

  4. Methods to Solve • Graphing method • Substitution method • Linear combination/Elimination method

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

  6. 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

  7. 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.

  8. 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!

  9. 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

  10. 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

  11. 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

  12. 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

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