Solving Systems of Equations: Graphical and Algebraic Methods
Learn how to solve systems of equations graphically, algebraically, and using tables. Explore scenarios with linear systems and model situations. Practice finding solutions through graphing and point of intersection methods. Understand classification of systems - consistent, inconsistent, dependent, independent. Solve word problems involving systems of equations.
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Presentation Transcript
3-1 Linear Systems Unit Objectives Solve a system of equations graphically and algebraically. Solve a system of linear inequalities. Model situations with linear systems Today s Objective: I can solve a system of equations by graphing and using tables.
If the sharks growth rate stays the same, at what age would they be the same length and how long will they be?
Solving a System using a graph: System of Equations: Solution of a System: Points that make all the equations true. 1. Graph the equations - change to y = mx + b - graph the intercepts 2. Find the point of intersection Calculator: 1. Put equations in [y =] 2. Adjust [window] to see intersection. 3. [2nd], [trace], [5] (intersection) 4. [enter], [enter], [enter] Two or more equations ? =3 3? + 2? = 8 3? + 2? = 4 2? + 4 ? = 3 2? 2 ( 2,1)
Solving a System using a graph: 1. Graph the equations 2. Find the point of intersection Calculator: 1. Put equations in [y =] 2. Adjust [window] to see intersection. 3. [2nd], [trace], [5] (intersection) 4. [enter], [enter], [enter] Table: 1. Put equations in y = 2. [2nd], [graph] 3. Scroll to matching y values ? =1 2? 2 ? 2? = 4 3? + ? = 5 ? = 3? + 5 (2, 1)
Classifying a system Consistent: has a solution Inconsistent: has no solution Dependent: infinite solutions Independent: one solution Same slopes Different y-intercepts Same slopes Same y-intercepts Different slopes Inconsistent 2? + 4? = 6 4? + 8? = 12= ? = 0.5? + 1.5 ? = 0.5? 1.5
x = years y = length ? = 0.75? + 37 ? = 1.5? + 22 If the sharks growth rate stays the same, at what age would they be same length and how long will they be? 20 years and they will be 52 cm long 3-1 p.138:7-13 odd, 17-37 odd Graph on graph paper, then check on calculator.
System of Equations A local gym offers two monthly membership plans. Visits v Plan A 100 + 5v 200 Plan B 10v 200 Plan A: One-time sign-up fee of $100 and charges $5 each time you use the gym. 20 Plan B: No sign-up fee but charges $10 each time you use the gym. When will the two plans cost the same?
Example 2 Josie makes and sells silver earrings. She rented a booth at a weekend art fair for $325. The materials for each pair of earrings cost $6.75, and she sells each pair for $23. 1. Write two equations to model the cost and income for Josie. 2. Create a table and graph to find the solution. 3. What does this solution mean? x = Number of earrings y = Dollars made or spent The solution is where Josie will break even. Earrings x 20 Cost Income 23x 460 325 + 6.75x 460
Example 3 Edna leaves the trailhead to hike 12 miles toward the lake. Maria leaves the lake to hike towards the trailhead. Edna walks uphill at 1.5 miles/hour, while Maria walks downhill at 2.5 miles/hour. 1. write two equations to model both girls time and distance they are hiking. (Be sure define your variables) Create a table and graph to find the solution. What does this solution mean? t = time spent hiking d = distance traveled from trailhead The solution is where Edna and Maria will meet. time Edna 1.5t 4.5 Maria 12 2.5t 4.5 t 3
