3. Graphing Systems of Equations Graph of a System Intersecting lines- intersect at one point One solution Same Line- always are on top of each other, slope is equal. Infinitely many solutions Parallel lines- are opposite from each other and will never! intersect. No solution

4. Example 1-Number of solutions Y=-x+5 Y=x-3 Since the graph of y=-x=5 and y=x-3 are intersecting lines there is one solution. Example 2-Solve a System of Equations Y=-x+8 y=4x-7 the graph appears to intersect at the point with coordinates (3,5). Check the estimate by replacing x with 3 and 5 with y in both equations.

5. Substitution Example 1-Solve using Substitution Y=3x X+2y=-21 Since y =3x, substitute 3x for y in the second equation. X+2y=-21 2 nd equation X+2(3x)=-21 y=3x x+6x=-21 simplify 7x=-21 combine like terms 7x/7=-21/7 divide each side by 7 X=-3 Use y=3x to find the value of y. y=3x y=-9 Y=3(-3) The solution is ( -3,-9)

6. Elimination Using Addition and Subtraction Example 1-Elimination using Addition 3x-5y=-16 2x+5y=31 Since the coefficients of the y terms,-5 and 5, are additive inverse, you can eliminate the y terms by adding the equations. 3x-5y=-16 +2x+5y=31 Notice the y variable is eliminated. 5x = 15 5 5 divide each side by 5 x=3 simplify

7. Example 1-Continued Now substitute 3 for x in either equation to find the value of y. 3x-5y=-16 first equation 3(3)-5y=-16 replace x with 3 9-5y=-16 simplify -9 -9 subtract 9 from each side -5y=-25 simplify -5 -5 divide each side by -5 Y=5 The solution is (3, 5)

8. Example 2-Elimination Using Subtraction 5s+2t=6 9s+2t=22 Since the coefficients of the t terms, 2 and 2, are the same you can eliminate them by using subtraction. 5s+2t=6 (-)9s+2t=22 Notice that the variable t is eliminated. -4s -16 -4 -4 Divide each side by -4 S=4 Simplify

9. Example 2- Continued Now substitute 4 for s in either equation to find the value of y. 5s+2t=6 5(4)=2t=6 Replace s with 4 20 +2t=6 Simplify -20 -20 Subtract 20 from each side 2t=-14 Simplify 2 2 Divide each side by 2 t=-7 The solution is (4, -7)

10. Elimination Using Multiplication Example 1-Multiply one Equation to Eliminate 3x+4y=6 5x+2y=-4 Multiply the 2 nd equation by -2 so the coefficients of the y terms are additive inverse. Then add the equations. 3x+4y=6 5x+2y=-4 (+)-10x-4y=8 Mult by 2 -7x 14 Add the equations -7 -7 Divide each side by -7 x=-2

11. Example 1- Continued Now substitute -2 for x in either equation. 3x+4y=6 3(-2)+4y=6 x=-2 -6+4y=6 Simplify +6 +6 add 6 to both sides 4y=12 Simplify 4 4 Divide by 4 on each side y=3 The solution is (-2, 3)