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Ch 1.4: Solving Linear Systems

Learn how to determine the number of solutions in linear systems using graphing, substitution, and elimination methods, with practical examples and practice tests provided.

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Ch 1.4: Solving Linear Systems

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  1. Ch 1.4: Solving Linear Systems Essential Question: How can you determine the number of solutions of a linear system?

  2. Warm Up During one season, a police officer issued a total of 375 citations for warning and speeding. Of these, there were 37 more warnings than speeding tickets. How many warnings and how many speeding tickets were issued? Let w = number of warnings Let s = number of speeding tickets w + s = 375 w = s + 37

  3. What is a system of equations? • A system of equations is two or more equations in two or more variables. • A solution of this system is an ordered pair that satisfies each equation in the system. 2x + y = 5 3x – 2y = 4 • Which ordered pair satisfies both equations? (4,3) (2,1) or (5,6)

  4. Method of Graphing When you use this method, the solution is the point of intersection because that is where the (x,y) values are equal. To use this method, • Graph each equation ( it might be helpful to solve for y in terms of x 2) Find the intersection point 2x + y = 5 3x – 2y = 4

  5. Method of Substitution To use this method, 1) Solve one of the equations in terms of the other 2) Substitute the expression found in step 1 into the other equation to obtain an equation in one variable 3) Solve the equation obtained in Step 2. 4) Back substitute the values obtained in Step 3 into the expression obtained in step 1 to find the values of the other variable 5) Check that each solution satisfies both of the original equations 2x + y = 5 3x – 2y = 4

  6. Method of Elimination • Obtain coefficients for x (or y) that differ only in sign by multiplying all terms of one of both equations by suitable chosen constants • Add the equations together. • Back substitute the value obtained in step 2 into either of the original equations and solve for the other variable • Check your solution in both of the original equations 2x + y = 5 3x – 2y = 4 Solve the Original Example with this method!

  7. A gym offers two options for membership plans. Option A includes an initiation fee $121 and costs $1 per day. Option B has no initiation fee but costs $12 per day. After how many days will the total costs of the gym membership plan be equal?

  8. Ch 1.4: Solving Linear Systems Essential Question: How can you determine the number of solutions of a linear system? Topic/Question Notes A Linear equation in 3 variables x, y, and z An equation of the form ax + by + cz = d where a, b, and c are NOT ALL zero.

  9. Visualizing Solutions of Systems

  10. EX 0: x – 2y + 3z = 9 y + 4z = 7 z = 2

  11. Practice Test

  12. quadratic quadratic Absolute value linear cubic Square root

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