Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations are different Equations are the same Slopes are the same 1
Solving Systems by Graphing System of Linear Equations two or more linear equations Solving a System of Equations by Graphing STEP 1 Both equations must be in slopeintercept form (y = mx + b) STEP 2 Graph each equation. You need to plot as many points possible to get the most accurate line. STEP 3 Determine the point where the lines intersect. STEP 4 Check the point by substituting into both equations 2
Example Solve each system by graphing. y = x + 4 y = x + 2 Example Solve each system by graphing. x + y = 3 x 2y = 0 3
Solving Systems of Equations by Substitution Step 1 Solve one equation for one variable Step 2 Substitute for the variable from Step 1 into the other equation. Step 3 Substitute for the variable from Step 2 into any of the original two equations. 4
Example Solve each system by substitution. y = 2x x + 2y = 15 5
Example Solve each system by substitution. x + y = 3 x = 2y 6
Example Solve each system by substitution. 2x = 2y + 6 2x 3y = 6 7
Solving Systems of Equations by Elimination Step 1 Align the like terms Step 2 Eliminate one of the variables. Solve for the remaining variable. Step 3 Substitute for the variable from Step 2 into any of the original two equations. 8
Example Solve each system by elimination. x + 2y = 9 x + 3y = 16 9
Example Solve each system by elimination. 2x + y = 9 3x y = 16 10
Example Solve each system by elimination. 3x + 5y = 7 10x + 5y = 0 11
Example Solve each system by elimination. 3x + y = 7 2x 2y = 6 12
Example Solve each system by elimination. 3x + 2y = 1 4x + 3y = 2 13
Graphing Linear Inequalities Linear inequality resembles a linear equation, but with an inequality symbol instead of an equal sign. Step 1 Write your equation in slope intercept form ( y = mx + b). Must follow inequality rules! Step 2 Graph (Use m and b) Step 3 Determine whether your line is dashed or solid. or solid < or > dashed Step 4 Shade appropriate side. > or shade upper part < or shade lower part 14
Example Graph the linear inequality y > 3x + 5 15
Example Graph the linear inequality y 2x 3 16
Example Graph the linear inequality 3y > 2x + 9 17
Example Graph the linear inequality 5x 2y < 10 18
Example Graph the linear inequality x 2y 8 19
Example Graph the linear inequality y 2 20
Example Graph the linear inequality x > 1 21
Systems of Inequalities A set of two or more inequalities. Step 1 Each equation has to be in slope intercept form Step 2 Graph each line. (Use slope and y intercept). Step 3 Determine solid or dashed line and shade appropriate side. 22
Example Solve each system of inequalities by graphing. y > 2x + 3 y 3x + 8 23
Example Solve each system of inequalities by graphing. y 3x 6 m = 3 1 b = 6 x + y < 6 x x y < x + 6 m = 1 1 b = 6 24
Example Solve each system of inequalities by graphing. 6x + 3y < 6 6x 6x 3y < 6x + 6 3 3 y < 2x + 2 m = 2 1 b = 2 x + y 2 +x +x y x + 2 m = 1 1 b = 2 25