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