Warm Up 1. Graph y = 2x – 3 2. Graph y = ½ x + 2 3. Graph 6x + 3y = 9 Have students choose their own x values. Discuss why the graph is still the same even though they chose different values. Have them plot someone else’s points on their graph as well.

Types of Systems There are 3 different types of systems of linear equations 3 Different Systems: Infinite Solutions No Solution One solution

Type 1: Infinite Solutions A system of linear equations having an infinite number of solutions is described as being consistent-dependent. y  The system has infinite solutions, the lines are identical x

Type 2: No Solutions Remember, parallel lines have the same slope A system of linear equations having no solutions is described as being inconsistent. y  The system has no solution, the lines are parallel x Remember, parallel lines have the same slope

Type 3: One solution A system of linear equations having exactly one solution is described as being one solution. y  The system has exactly one solution at the point of intersection x

So basically…. If the lines have the same y-intercept b, and the same slope m, then the system has Infinite Solutions. If the lines have the same slope m, but different y-intercepts b, the system has No Solution. If the lines have different slopes m, the system has One Solution.

Solve Systems of Equations by Graphing

Steps Make sure each equation is in slope-intercept form: y = mx + b. Graph each equation on the same graph paper. The point where the lines intersect is the solution. If they don’t intersect then there’s no solution. Check your solution algebraically.

Graph to find the solution. y = 3x – 12 y = -2x + 3 Solution: (3, -3)

1. Graph to find the solution.

2. Graph to find the solution. No Solution

3. Graph to find the solution.

4. Graph to find the solution.

Solve Systems of Equations by Elimination

Analyzing the Solution Does the solution still work if you multiply one or both of the equations by a number like 2 or -2?

Steps for Elimination: Arrange the equations with like terms in columns Multiply, if necessary, to create opposite coefficients for one variable. Add/Subtract the equations. Substitute the value to solve for the other variable. Write your answer as an ordered pair. Check your answer.

EXAMPLE 1 (continued) (-1, 3)

EXAMPLE 2 4x + 3y = 16 2x – 3y = 8 (4, 0)

EXAMPLE 3 3x + 2y = 7 -3x + 4y = 5 (1, 2)

EXAMPLE 4 2x – 3y = -2 -4x + 5y = 2 (2, 2)

EXAMPLE 5 5x + 2y = 7 -4x + y = –16 (3, -4)

EXAMPLE 6 2x + 3y = 1 4x – 2y = 10 (2, -1)

Classwork (-1, 4) (-1, 2) (-2, 2) (1, 5) (5, -2) (4, -1) (-1/2, 4) Add/Subtract Use elimination to solve each system of equations. 1. -6x – 5y = -4 2. 3m – 4n = -14 3. 3a + b = 1 6x – 7y = -20 3m + 2n = -2 a + b = 3 -3x – 4y = -23 5. x – 3y = 11 6. x – 2y = 6 -3x + y = 2 2x – 3y = 16 x + y = 3 2a – 3b = -13 8. 4x + 2y = 6 9. 5x – y = 6 2a + 2b = 7 4x + 4y = 10 5x + 2y = 3 (-1, 4) (-1, 2) (-2, 2) (1, 5) (5, -2) (4, -1) (-1/2, 4) (1/2, 2) (1, -1)

Classwork (1, 1) (-3, 4) (-1, 2) (-1, 2) (4, -2) (-4, -2) (2, -1) Multiply Use elimination to solve each system of equations. 2x + 3y = 6 2. 2m + 3n = 4 3. 3a - b = 2 x + 2y = 5 -m + 2n = 5 a + 2b = 3 4x + 5y = 6 5. 4x – 3y = 22 6. 3x – 4y = -4 6x - 7y = -20 2x – y = 10 x + 3y = -10 4x – y = 9 8. 4a – 3b = -8 9. 2x + 2y = 5 5x + 2y = 8 2a + 2b = 3 4x - 4y = 10 (1, 1) (-3, 4) (-1, 2) (-1, 2) (4, -2) (-4, -2) (2, -1) (-1/2, 2) (2.5, 0)

Solve Systems of Equations by Substitution

Steps One equation will have either x or y by itself, or can be solved for x or y easily. Substitute the expression from Step 1 into the other equation and solve for the other variable. Substitute the value from Step 2 into the equation from Step 1 and solve. Your solution is the ordered pair formed by x & y. Check the solution in each of the original equations.

Solve by Substitution 1. x = -4 3x + 2y = 20 1. (-4, 16)

Solve by Substitution 2. y = x - 1 x + y = 3 2. (2, 1)

Solve by Substitution 3. 3x + 2y = -12 y = x - 1 3. (-2, -3)

Solve by Substitution 4. x = -5y + 4 3x + 15y = -1 5. No solution

Solve by Substitution 5. x = 5y + 10 2x - 10y = 20 7. Many solutions

CW/HW