Solving Systems of Equations by Graphing What is a System of Equations? Solving Linear Systems – The Graphing Method Consistent Systems – one point (x,y) solution Inconsistent Systems – no solution Dependant Systems – infinite solutions Solving Equations Graphically

Concept: A System of Linear Equations Any pair of Linear Equations can be a System A Solution Point is an ordered pair (x,y) whose values make both equations true When plotted on the same graph, the solution is the point where the lines cross (intersection) Some systems do not have a solution

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

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 skateboards f(x) = 15000 + 12.5x (total cost equation) g(x) = 50x (total revenue equation)

Using Algebra to Check a Proposed Solution Is (3,0) also a solution?

Estimating a Solution using The Graphing Method Graph both equations on the same graph paper If the lines do not intersect, there is no solution If they intersect: Estimate the coordinates of the intersection point Substitute the x and y values from the (x,y) point into both original equations to see if they remain true equations

Approximation … Solving Systems Graphically

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 true 2 + 1 = 3 true therefore, (1,2) is the solution (1,2)

Practice – Solving by Graphing Inconsistent: no solutions y = -3x + 5  (0,5) and (3,-4) y = -3x – 2  (0,-2) and (-2,4) They look parallel: No solution Check it: m1 = m2 = -3 Slopes are equal therefore it’s an inconsistent system

Practice – Solving by Graphing Consistent: infinite sol’s 3y – 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 -4 and it becomes identical to the 1st equation therefore, consistent, dependant system (1,2)

The Downside of Solving by Graphing: It is not Precise

Summary Solve Systems by Graphing Them Together Graph neatly both lines using x & y intercepts Solution = Point of Intersection (2 Straight Lines) Check by substituting the solution into all equations Cost 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) Graphing can solve equations having one variable

Practice Problems Page 112 Problems 4-9, and 13-36