2 Section 3.1 Solving Systems using tables and graphs What is a System of Equations?Solving Linear Systems – The Graphing MethodConsistent Systems – one point (x,y) solutionInconsistent Systems – no solutionDependant Systems – infinite solutionsSolving Equations Graphically
3 Concept: A System of Linear Equations Any pair of Linear Equations can be a SystemA Solution Point is an ordered pair (x,y) whose values make both equations trueWhen plotted on the same graph, the solution is the point where the lines cross (intersection)Some systems do not have a solution
4 Why Study Systems of Equations? We will study systems of 2 equations in 2 unknowns (usually x and y)The algebraic methods we use to solve them will also be useful in higher degree systems that involve quadratic equations or systems of 3 equations in 3 unknowns
5 A “Break Even Point” Example. A $50 skateboard costs $12. 50 to build, A “Break Even Point” Example A $50 skateboard costs $12.50 to build, once $15,000 is spent to set up the factory:Let x = the number of skateboardsf(x) = x (total cost equation)g(x) = 50x (total revenue equation)
6 Using Algebra to Check a Proposed Solution Is (3,0) also a solution?
7 Estimating a Solution using The Graphing Method Graph both equations on the same graph paperIf the lines do not intersect, there is no solutionIf they intersect:Estimate the coordinates of the intersection pointSubstitute the x and y values from the (x,y) point into both original equations to see if they remain true equations
9 Practice – Solving by Graphing Consistent: (1,2)y – x = 1 (0,1) and (-1,0)y + x = 3 (0,3) and (3,0)Solution is probably (1,2) …Check it:2 – 1 = 1 true2 + 1 = 3 truetherefore, (1,2) is the solution(1,2)
10 Practice – Solving by Graphing Inconsistent: no solutionsy = -3x (0,5) and (3,-4)y = -3x – 2 (0,-2) and (-2,4)They look parallel: No solutionCheck it:m1 = m2 = -3Slopes are equaltherefore it’s an inconsistent system
11 Practice – Solving by Graphing Consistent: infinite sol’s3y – 2x = 6 (0,2) and (-3,0)-12y + 8x = -24 (0,2) and (-3,0)Looks like a dependant system …Check it:divide all terms in the 2nd equation by -4and it becomes identical to the 1st equationtherefore, consistent, dependant system(1,2)
12 Summary of Section 3.1 Solve Systems by Graphing Them Together Graph neatly both lines using x & y interceptsSolution = Point of Intersection (2 Straight Lines)Check by substituting the solution into all equationsCost and Revenue lines cross at “Break Even Point”A Consistent System has one solution (x,y)An Inconsistent System has no solution The lines are Parallel (have same slope, different y-intercept)A Dependent System happens when both equations have the same graph (the lines have same slope and y-intercept)
13 Homework pgHomework: P. 138 #7,9,13,14,29,31,33,38,39,40,41,53,54 Challenge: #48
14 What Next?Present Section 3.2 Solving Systems Algebraically
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