7.1 Graphing Linear Systems Systems of Equations: A solution to a system of equations in two variables x and y is an ordered pair (x,y) of real numbers that satisfy each equation in the system.

7.1 Graphing Linear Systems Solve the following system of equations: y = -2x + 6 y = ½x + 1

Systems of Equations Solve the following system of equations: y = -2x + 6 y = ½x + 1

Systems of Equations Example 1. Solve the following system of equations: y = -2x + 6 y = ½x + 1

7.1 Graphing Linear Systems Solve the following system of equations: y = -2x + 6 y = ½ + 1

7.1 Graphing Linear Systems Solve the following system of equations: y = -2x + 6 y = ½x + 1 Solution: (2, 2)

7.1 Graphing Linear Systems Solve the following system of equations: y = 2x + 1 y = ½x + 3

Systems of Equations Solve the following system of equations: y = 2x + 1 y = ½x + 3

7.1 Graphing Linear Systems Solve the following system of equations: y = 2x + 1 y = ½x + 3

7.1 Graphing Linear Systems Solve the following system of equations: y = 2x + 1 y = ½x + 3

7.1 Graphing Linear Systems Solve the following system of equations: y = 2x + 1 y = ½x + 3 Solution: (4/3, 11/3)

7.1 Graphing Linear Systems Example: x + y = 4 2x – y = 5 3 + 1 = 4 2(3) – 1 = 5 The point (3,1) is a solution to the system of equations.

7.1 Graphing Linear Systems Solving by Graphing x + y = 4 2x – y = 5 (3,1) Put in slope-intercept form y = -x + 4 y = 2x - 5

7.1 Graphing Linear Systems Solving by Graphing y = x - 2 y = -2x + 1 (1,-1) 14

7.1 Graphing Linear Systems Solving by Graphing x + y = 4 2x + 3y = 6 Put in slope-intercept form (6,-2) y = -x + 4

7.2 Substitution Solving by Substitution x + y = 4 2x – y = 5 2x – (-x + 4) = 5 y = 1 2x + x – 4 = 5 Solve one of the equations for x or y and substitute into the other equation. (3,1) 3x – 4 = 5 3x = 9 x = 3

7.2 Substitution Solving by Substitution x + y = 4 2x + 3y = 6 2x +3(-x + 4) = 6 y = -2 Solve one of the equations for x or y and substitute into the other equation. 2x – 3x + 12 = 6 (6,-2) -x + 12 = 6 -x = -6 x = 6

7.3 Elimination Solving by Elimination x + y = 4 2x – y = 5 Add to eliminate y 3x = 9 x = 3 y = 1 (3,1) Remember to substitute back in to find the other variable

7.3 Elimination Solving by Elimination x + y = 4 2x + 3y = 6 Multiply the top equation by -3 -3x – 3y = -12 2x + 3y = 6 (6,-2) -x = -6 x = 6 y = -2

7.5 Special Types of Systems If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. y Solution x

7.5 Special Types of Systems If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. x

7.5 Special Types of Systems If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. y x