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Lesson 23a Systems of equations by substitution.notebook
Do Now: Solve the following systems of equations graphically
2x + 3y = -3
y = 3
y = -2 x - 1 3
2x + 3y = -3
y = 3
Solution = (-6,3)
Do Now: Solve the following systems of equations graphically
2x + 3y = -3
y = 3
Can you think of a way to solve this system using aSUBSTITUTION METHOD?
Lesson 23a Systems of equations by substitution.notebook
Aim: Substitution method for Solving Systems of Equations
Grab and Plug
2x + 3y = -3y = 3
2x + 3y = -3
y = 3
2x + 3(3) = -32x = -12 x = -6
The solution point is ( -6, 3) or x =-6 and y = 3
ex 1
Lesson 23a Systems of equations by substitution.notebook
(Substitution Method)
x = 2y - 52x + 3y = 11
1) Solve for one of the variables
2) Substitute the expression in for the variable in the other equation
3) Solve for x and y
5) Check in both original equations
ex 2
2(2y 5) + 3y = 114y 10 + 3y = 11 7y = 21
y = 3 x = 2y 5x = 2(3) 5x = 1
Solution: (1,3)
Example 3
y = 2x - 1
y = x - 1
5) Check in both original equations
4) Find the value of the other variable by using the number you obtained in steps 1-3
3) Solve for the variable
2) Substitute the expression in for the variable in the other equation
1) Solve for one of the variables
x 1 = 2x 1 1 = x 1 0 = x
y = x 1y = 0 1y = 1
Solution: (0,1)
Lesson 23a Systems of equations by substitution.notebook
x = 3y - 1
-2x = y + 9
Example 4
5) Check in both original equations
4) Find the value of the other variable by using the number you obtained in steps 1-3
3) Solve for the variable
2) Substitute the expression in for the variable in the other equation
1) Solve for one of the variables
2(3y 1) = y + 96y + 2 = y + 9 7y + 2 = 9 7y = 7 y = 1
x = 3y 1 x = 3(1) 1 x = 3 1 x = 4
Solution: (4, 1)
Special Cases
x + 3y = 8
x = 5 - 3y
When the answer turns out to be a contradiction, there is nosolution. The two lines are parallel.
5 3y + 3y = 8 5 = 8
No Solution
Lesson 23a Systems of equations by substitution.notebook
Special Cases
x + 3y = 8
x = 5 - 3y
y = -1 x + 8 3 3
y = -1 x + 5 3 3
x + 3y = 8x = 5 - 3y
2x - 4y = -2
-x + 2y = 1
Special Cases
When the answer is an identity, the solution is the equation itself. We say it is all real numbers (infinite solutions).
x = 2y 1Solve for x.
2(2y 1) 4y = 24y 2 4y = 2 2 = 2
Infinite Solutions
Lesson 23a Systems of equations by substitution.notebook
2x - 4y = -2
-x + 2y = 1
Special Cases
y = 1 x + 1 2 2
y = 1 x + 1 2 2
-x + 2y = 1
2x - 4y = -2
Using the Substitution Method to solve systems of Linear Equations
In the substitution method a quantity may be substituted for its equal.
Ex. 4x+3y=27 and y = 2x 14x + 3 (2x1) =274x +6x 3 =2710x 3 = 2710x = 30x=3
Replace the new "x" value in either of the original equations involving both variables and solve for "y"
y=2x1y=2(3)1y=61y=5
Solution (3,5)
Lesson 23a Systems of equations by substitution.notebook
Check: Substitute the 3 in for x and the 5 in for y in each of the original equations. The ordered pair must be true for both equations
4x +3y=27 y=2x14(3)+3(5) = 27 5=2(3)112+15 = 27 5=6127 = 27 5=5
solve the following systems of equations using the substitution method. Show all work.
1. y = 2x and x+y=21
2. 3y2x=11 and y=2x+9
Solution (7,14)
Solution (2,5)
Lesson 23a Systems of equations by substitution.notebook
3. 7x3y=23 and x = 2y+13
4. y=3x5 and y12=3x
Solution (5,4)
No solution Lines are parallel
