Solving by Substitution

(4,3)

2x − 3y = −1 y = x − 1

Infinitely Many Solutions

3x + 4y = 8 y = -3/4x +2

One Solution Equations

Characteristics: Only one point (x,y) Different slopes Different y-intercepts Independent and Consistent

No Solution Equations

Characteristics: Parallel Lines Same Slope Different y-intercept Dependent and Inconsistent

Infinitely Many Solution Equations

Characteristics: Same Slope Same y-intercept Dependent and Consistent

y

Given this system of equations, after substituting in step 2 you will solve for which variable? x = 8y + 5 2x + 3y = 5

x

Given this system of equations, after substituting in step 2 you will solve for which variable? y = 2x 4x + 5y = 8

10y+3=7

Given this system of equations, after substituting in step 2, simplify the left side of the remaining equation. Do not solve. x = 3y + 1 3x + y = 7

(-2,3)

Given this system of equations, after substituting, simplify the left side of the remaining equation. Do not solve. 2.5x + y = -2 3x + 2y = 0

9x

Given this system of equations, after substituting, simplify the left side of the remaining equation. Do not solve. y = 2x 5x + 2y = 12

y + 3 + y = 8

Given this system, when rewriting the equation, immediately after substituting in step 2, without any simplifying, what does the equation look like. x = y + 3 x + y = 8

2(5x) + 3x = 26

Given this system, when rewriting the equation, immediately after substituting in step 2, without any simplifying, what does the equation look like. y = 5x 2y + 3x = 26

Practice Graphing

Graph these equations 1) x - y = -2 2x + y = 8 Ans: (2,4) How'd I get this? Graph it and see!

Practice Graphing

Graph these equations 2) -3x + y = 1 4x + y = 8 Ans: (1,4) How'd I get this? Graph it and see!

-2x - 3y = 12 ; -4x - 6y = 24

Infinitely many solutions

3x - 3y = -15 ; -x + y = 5

Infinitely many solutions

x - 8y = 17 ;- 3x + 24y = -51

Infinitely many solutions

5x - y = 18 ; 10x - 2y = 32

No Solution

3x+4y=36 ; 6x+8y=48

No solution

x + y = 3 ; -4x - 4y = 12

No solution

Best Quarterback

Peyton Manning

What do you need to know in order to solve by graphing?

Slope and y-intercept

x + 2

What expression will you use for your substitution in step 2? (Take the easiest one!) 2y = 2x + 4 5y + 2x = 8

y + 7

What expression will you use for your substitution in step 2? 3y + 5x = 15 x = y + 7

2x + 3

What expression will you use for your substitution in step 2? y = 2x + 3 2y + 3x = 4

3x - 5

What expression will you use for your substitution in step 2? y = 3x - 5 y + 2x = 8

7x + 3

What expression will you use for your substitution in step 2? y = 7x + 3 2x + 8y = 19

(7, 3)

What is the solution to this system of equations. Solve by using substitution. x + 2y = 13 3x - 5y = 6

No Solution

What is the solution to this system of equations. Solve by using substitution. x + 5y = 4 3x + 15y = -1

No Solution

What is the solution to this system of equations. Solve by using substitution. x - 5y = 10 2x - 10y = -9

Infinitely Many

What is the solution to this system of equations. Solve by using substitution. x - 5y = 10 2x -3y = -9

(6,-3)

What is the solution to this system of equations. Solve by using substitution. 2x - 3y = 21 y = 3 - x

(6,-3)

What is the solution to this system of equations. Solve by using substitution. 2x - 3y = 21 y= 3 - x

(8,-2)

What is the solution to this system of equations. Solve by using substitution. x = -4y 3x + 2y = 20

(16,4)

What is the solution to this system of equations. Solve by using substitution. x = 4y 3x - 2y = 40

(-6,1)

What is the solution to this system of equations. Solve by using substitution. x = y - 7 x + 8y = 2

(8,-2)

What is the solution to this system of equations. Solve by using substitution. x= -4y 3x + 2y =20

(-2,-4)

What is the solution to this system of equations. Solve by using substitution. y = 2x x + 3y = -14

(-1,5)

What is the solution to this system of equations. Solve by using substitution. y = 3x + 8 5x +2y = 5

(1,4)

What is the solution to this system of equations. Solve by using substitution. y = 4x x + y = 5

(-2, -9)

What is the solution to this system of equations. Solve by using substitution. y = 4x -1 y = 2x -5

(-2, -9)

What is the solution to this system of equations. Solve by using substitution. y = 4x -1 y= 2x -5

(3,7)

What is the solution to this system of equations. Solve by using substitution. y = 5x - 8 4x + 3y = 33

(5,8)

What is the solution to this system of equations. Solve by using substitution. y = x + 3 2x + y = 18

(2,1)

What is the solution to this system of equations. Solve by using substitution. y = x -1 x + y = 3

(2,1)

What is the solution to this system of equations. Solve by using substitution. y= x -1 x + y =3

(3,7)

What is the solution to this system of equations. Solve by using substitution. y=5x - 8 4x + 3y = 33

1

What is the x value of the solution to this system of equations when solving by substitution x = 2y - 5 2x + 3y = 11

3

What is the y value of the solution to this system of equations when solving by substitution x = 2y - 5 2x + 3y = 11

(2,3)

y = 5x − 7 −3x − 2y = −12

2x + 6y = 16 ; 3x - 5y = -18

(-1,3)

y=-x-3 y=x+1

(-2,-1)

y=3x+4 y=-x-4

(-2,-2)

x - 3y = 10 ; x + 5y = -22

(-2,-4)

y=2x+3 y=x-1

(-4,-5)

y=x-4 y=2x

(-4,-8)

No Solution

y = 2x - 5 -2x + y = 8

y=x-6 y=2x

(-6,-12)

y=-x y=-7x

(0,0)

y=-2x+4 y=-x+3

(1,2)

5x + y = 8 ; 2x + y = 5

(1,3)

y=2x+5 y=-x+8

(1,7)

y=-x-14 y=-8x

(2,-16)

7x + 2y = 24; -6x + 3y = 3

(2,5)

x + 7y = -11 ; -2x = 5y = 4

(3,-2)

y=x+20 y=6x

(4,24)

5x - 3y = 22 ; -4x + y = -19

(5,1)

Steps: 1) Pick one equation and solve for one of the variables. 2) Substitute the answer from the 1st step, which in this case is y = -2x + 14, into the other equation and solve for the 2nd variable. 3) Now that you know a single number for one variable, substitute it into one of the original equations and find the 2nd variable number. 4) Write your two variable answers into an ordered pair (x,y).

ex: 4x + 2y = 8 Solve for either x or y in one of the x + y = 14 two equations. Step 1: x + y = 14 -x -x y = -x +14 Step 2: 4x + 2y = 8 4x + 2(-x +14) = 8 4x - 2x + 28 = 8 2x + 28 = 8 -28 -28 => 2x = -20 x = -10 Step 3: x + y = 14 -10 + y = 14 +10 +10 y = 24 Step 4: (-10, 24)

substitution

solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation

(2,1)

y = 6x − 11 −2x − 3y = −7

(3,-2)

y = −2 4x − 3y = 18

(1,2)

y = −3x + 5 5x − 4y = −3

How do you find slope?

y2-y1/x2-x1 or the difference of y divided by the difference of x

(-4,3)

−3x − 3y = 3 y = −5x − 17

(-3,-6)

−4x + y = 6 −5x − y = 21

(0,-2)

−5x + y = −2 −3x + 6y = −12

(0,-3)

−5x + y = −3 3x − 8y = 24

(3,-4)

−7x − 2y = −13 x − 2y = 11