Solving Systems of Equations by Graphing Chapter 3.1

System of Equations 2 or more equations with the same variables One way of solving: – Graph all equations – The solution is where the graphs intersect

Example 1 y = ½ x y = -x + 6 Both equations are in slope-intercept form, so they are easy to graph. They intersect at (4, 2) y = ½ x y = -x + 6

Example 2 A service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200 and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? Can we represent this situation with a system of equations? – Cost:y = 2x – Income:y = 6x

Example 1 y = 2x y = 6x Both equations are in slope-intercept form, so they are easy to graph. They intersect at (50, 300) y = 2x y = 6x

Classifications of Systems Consistent: At least one solution – Independent: Exactly one solution – Dependent: Infinitely many solutions (same line) Inconsistent: No solutions (parallel lines)

Example 3 Graph the system of equations and describe as consistent/independent, consistent/dependent, or inconsistent. x – y = 5 x + 2y = -4 Equations are not in slope- intercept form, so let’s begin by rearranging them. x – y = 5 -x -y = -x + 5 y = x – 5 x + 2y = -4 -x 2y = -x – 4 y = - ½ x – 2 Now graph these lines

Example 3 y = x – 5 y = - ½ x – 2 The solution is (2, -3) The system is consistent and independent y = x - 5 y = - ½ x - 2

Example 4 9x - 6y = -6 6x - 4y = -4 We should begin by rearranging both equations into slope- intercept form Try this on your own Both equations simplify to y = 3/2 x + 1 Because lines overlap everywhere, there are infinitely many solutions This system is consistent and dependent y = 3/2 x + 1

Example 5 15x - 6y = 0 5x - 2y = 10 Rearrange these equations into slope-intercept form y = 5/2 x y = 5/2 x – 5 These lines are parallel, therefore they will never intersect The system is inconsistent y = 5/2 x y = 5/2 x -5