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Introduction to Systems of Equations (and Solving by Graphing)
Unit 5 Day 3
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Systems Two or more linear equations together form a system of linear equations. Example: A solution of the system of linear equations is any ordered pair that makes all the equations true. There are several ways to find the solution for a system of linear equations. Today, we will explore how to solve by graphing.
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Example One (-1, -2) none (1, 3)
Looking at these graphs of a system of equations, where do you think the solution is? Solution: _______ Solution: _______ Solution: _______ (-1, -2) none (1, 3)
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There are 3 types of solutions
ONE SOLUTION When the lines intersects at one point. (x, y)
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There are 3 types of solutions
NO SOLUTION When the lines are parallel and never intersect
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There are 3 types of solutions
INFINITELY MANY SOLUTIONS When the two equations graph the same exact line.
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Example Two a.) y = -1/2x + 2 3x + y = -3 y = -3 – 3x
(subtract 3x to solve for y) y = -3 – 3x Solution: ( -2 , 3 )
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Example Two: b.) y = 2x + 1 -12x + 6y = 6 (add 12x to both sides)
(divide by 6 to both sides) y = 1 + 2x SAME LINE! Infinitely many solutions
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Example Two: c.) y = x + 5 y = -4x
Let’s check your answer by calculator (watch this video to see it done by a person): “Y=” to put in your equations Press “GRAPH” (ZoomOut if necessary) CALC menu (2nd “TRACE”) Option 5: intersect ENTER : 3 times SOLUTION: (-1, 4)
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Example Three This graph shows the money charged given number of minutes used. How much does Company S charge as a starting fee? _____ How much does Company T charge as a starting fee? _____ $15 $5
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Example Three This graph shows the money charged given number of minutes used. For how many minutes do the companies charge the same amount? ____ What is the charge for this? ____ 20 minutes $20
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Example Three This graph shows the money charged given number of minutes used. Which company charges more per minute? _________ Company S price per minute: ___________ Company T price per minute: ___________ +20 +20 +$5 +$15 Company T 5/20 = $0.25 per minute 15/20 = $0.75 per minute
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Example Three This graph shows the money charged given number of minutes used. Write an equation for Company S:______________ Company T: ______________ When would it be better to use Company T instead of Company S? y = $0.25x + 15 y = $0.75x + 5 When you use less than 20 minutes - because over 20 minutes, T is more expensive
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