Warm Up State the slope and the y-intercept for each of the equations below:

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1 Warm Up State the slope and the y-intercept for each of the equations below: 1) y = 3x + 2 2) y = -2x + 6 3) 2x + 3y = 15 4) 6y - 5x = -12

2 A SYSTEM OF EQUATIONS is a set of two or more equations that contain two or more variables. A SOLUTION OF A SYSTEM OF EQUATIONS is the point of intersection of the lines or can be said to be a set of values that are solutions of all the equations.

3 If you can graph a straight line, you can solve systems of equations graphically! Solving a system by graphing is very easy!!! To solve a system of equations graphically, just graph both equations and see where they intersect (cross). The point of intersection is the solution. FOR EXAMPLE: 4x + y = 8 y - x = 3 Graph: y = -4x + 8 y = x + 3 Point of intersection is:

4 Solutions to Systems of Equations There are 3 possible solutions for a system of equations: 1) 1 solution - this happens when the graphs of the equations intersect at exactly 1 point. y =x 4 y= 2x +5 Called consistent and independent. Notice the lines have different slopes! Solutions to Systems of Equations 2) No solution - this happens when the graphs of the equations never intersect, which means they are parallel lines. y = -x + 3 y = -x - 1 Called inconsistent. Notice the lines have the same slope BUT different y intercepts.

5 Solutions to Systems of Equations 3) Infinte solution - this happens when the graphs of the equations are the same line. This means they have the same slope and y- intercept. 2x + y = 3 4x + 2y = 6 Called consistent and dependent. Notice the lines have the same slope AND the same y intercepts. (Put into y=mx+b form) Solutions to Systems of Equations Description of Graph Number of Solutions Special Terminology Intersecting Lines Exactly One Consistent & Independent Parallel Lines NONE Inconsistent Same Line (Coincidental) Infinitely Many Consistent & Dependent

6 = Let's try another one: You try this one: =

7 Practice on Solving Systems by Graphing Solve each system by graphing: 1) y = 3x -4 y = -3x + 2 2) y = 4/3 x + 3 y = 2/3 x 3 3) y = 5/4 x -2 y = 5/4 x - 1 Practice on Solving Systems by Graphing Solve each system by graphing: 1) y = 3x -4 y = -3x + 2 2) y = 4/3 x + 3 y = 2/3 x 3 3) y = 5/4 x -2 y = 5/4 x - 1 ( 3, 1) no Solution (1, 1)

8 Graph the following and find the point of intersection, or solution ANSWERS 1) (-4, 7) 2) (3, -1) 3) (0, 1) 4) (3, 6) 5) (0, 2) 6) (0, -3)

9 Homework: Textbook pg 610 #1-6, 8, 10, 14, 16, 20 The substitution method is used to eliminate one of the variables by replacement when solving a system of equations! Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation.

10 STEPS FOR SOLVING A SYSTEM OF EQUATIONS BY SUBSTITUTION 1) Solve one of the equations for either "x =" or "y =". 2) Replace the "y" value in the first equation by what "y" now equals. Solve this new equation for "x". 3) Place this new "x" value into either of the ORIGINAL Pick the easier one to work with! 4) CHECK: Substitute your "x" and "y" values into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! FOR EXAMPLE: 3y - 2x = 11 y + 2x = 9

11 3x + 2y = 10 2x - y = 9 Let's try another system: 4x - y = 6 -x + y =3

12 You try this one: x + 4y = -10 x - 3y = 11 SPECIAL CASES We know the solution to a system is the intersection of two lines. Let's look at two special cases. CASE 1: y = 2x + 9 2x - y = -2 The solution to this system is

13 CASE 2: 10y = 2x - 8 x - 5y = 4 The solution to this system is Here is what you should remember! SYSTEMS OF EQUATIONS CAN HAVE... a POINT OF INTERSECTION which will be the answer to the system, no intersection (because they are parallel) and the answer to the system is NO SOLUTION, or many points of intersection (because they are the same line) and the answer to the system is INFINITELY MANY SOLUTIONS.

14 Look at the following examples: 6x - y = 12-12x + 2y = 4 x + y = 1-2y = 2x - 2 HOMEWORK Do the Pizzazz worksheet "Why Does the President Put Vegetables in His Blender?"

15 Simultaneous equations got you baffled? RELAX! You can do it! Think of the adding or subtracting method as simply "eliminating" one of the variables to make your life easier. STEPS FOR SOLVING A SYSTEM OF EQUATIONS BY ELIMINATION 1) First, "line" the variables up under one another. 2) Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be negatives of one another. 3) Now, subtract to eliminate the "x" or "y" variable. 4) Solve the simple equation. 5) Plug into either of the ORIGINAL equations to get the value of the other variable. Pick the easier one to work with! 6) CHECK: Substitute your "x" and "y" values into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE!

16 FOR EXAMPLE: 9x -3y = 3 3x + 8y = -17 Let's try another system: 2x + 4y = -4 3x + 5y = -3

17 Let's try another system: 5x + 2y = -1 3x + 7y = 11 You try this one: -2x + 5y = -17 3x - 10y = 28-5x + 2y = 32 2x + 3y = 10

18 You try this one: 3x + 2y = 6-6x - 4y = -12 2x - 4y = -1 6x - 12y = 0 HOMEWORK SOLVE EACH OF THE FOLLOWING SYSTEMS USING ELIMINATION OR LINEAR COMBINATIONS. Go to PAGE 610 and do 7, 10, 12, 19, 22, 23, 24 28, 30, and 34.

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