1. 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

2. 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

3. 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

4. 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)

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

6. 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

7. Approximation … Solving Systems Graphically

8. 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)

9. 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: m 1 = m 2 = -3 Slopes are equal therefore it’s an inconsistent system

10. 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 2 nd equation by -4 and it becomes identical to the 1 st equation therefore, consistent, dependant system (1,2)

11. Solving Equations by Graphing  Equations in one unknown can be split into two linear equations:  2x + 4 = -2  f(x) = 2x + 4 and g(x) = -2  When the two linear equations are graphed as a system, the solution is the point (-3,-2)  The x-coordinate is the solution to the original equation in one unknown!  The lines above cross at (-3,-2)  x = -3

12. The Downside of Solving by Graphing: It is not Precise

13. 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