DAY 2 – SUBSTITUTION
Objectives
Solve systems of equations using substitution .
Introduce solving systems by substitution. Explain that substitution means
replacing one
variable so that there is only one variable to solve for instead of two. Give
students an
example of substitution in real life . If I have 2 coke and 1 pepsi, then replace
the pepsi
with a coke, now I have only coke. The main thing the students need to
understand is
that by substituting, they end up with only 1 variable.
Model this method using algebra tiles . Use examples that solve for x first, that
is, y is
replaced.
Ex. 1 3x+y=8, y=x-4; since y is x-4, replace y in the first equation with x-4.
Using
parenthesis whenever anything is replaced is helpful when students need to
distribute so
they have it set up correctly. In this problem, 3x+(x-4)=8. (Solution: (3,-1))
Solve like a regular equation. Students can have the
equation mats and also a worksheet
where they solve the equation step by step . Use the value for x to solve for y
by
substituting x into either of the original equations.
Ex. 2 4x+3y=10 and y=x+1 (Solution: (1,2))
Use the algebra tiles to solve this problem. Write each step out on paper.
Sometimes, one equation is not solved in terms of the other variable. In these
problems,
first you must solve one equation so that it equals a variable. After one
equation is
solved, use the substitution method.
Ex. 3 4x+y=12, -2x-3y=14 (Solution: (5, -8))
In this problem, since the y in equation 1 is equal to one, it is easier to
solve this equation
in terms of y; y=-4x+12. Substitute this new value for y into equation 2 where y
is used.
DON’T forget the parenthesis!! -2x-3(-4x+12)=14. Solve for x. Use this value of
x to
solve for y.
Solve the following problems in student notebooks:
1. x=4y, 4x-y=75 (20, 5)
2. y=3x-8, y=4-x (3, 1)
3. x+3y=12, x-y=8 (9, 1)
4. x+14y=84, 2x-7y=-7 (14, 5)
5. 0.5x+4y=-1, x+2.5y=3.5 (6, -1)
6. At the end of the 2000 baseball season, the NY Yankees and the Cincinnati
Reds
had won a total of 31 World Series. The Yankees had won 5.2 times as many
World Series as the Reds. How many World Series did each team win?
(Y+R=31, Y=5.2R; y-26, r-5).
Take up these problems in class if time permits. Homework sheet is attached.
DAY 3 – ELIMINATION BY ADDITION/SUBTRACTION
Objectives
Solve systems of equations using elimination by addition and subtraction.
Introduce solving by elimination using addition or subtraction. Elimination
means
getting rid of something, in this case, remove a variable. Again, the idea is to
have only
one variable to solve for.
Given:
|
 |
Since the y values have a coefficients of
positive &
negative one , add them together to eliminate y. |
| |
|
Using x=37, substitute this new value for x into either of
the original equations.
Therefore, the solution is x=37 and y=25.
Given:
|
 |
Since the a values have coefficients of positive
one ,
subtract to eliminate a. |
| |
|
Using b=-12, substitute this new value for b into either
of the original equations.
Students try:
1) a-w=120; a+w=150 Ans – a=135, w=15
2) 4x+13y=40; 4x+3y=-40 Ans – x=-16, y=8
3) A resort hotel offers two weekend specials; Plan A is 3 nights with 6 meals
for $264,
Plan B is 3 nights with 2 meals for $218. What is the cost of one night’s
lodging and
what is the average cost per meal?
Solution:
The cost of a meal is $11.50.
Substitute m=11.5 into either original equation;
N =65 The cost for one night is $65.00.
Complete worksheet below in class or for homework.
DAY 4 – ELIMINATION BY MULTIPLICATION
Objectives
Solve systems of equations using elimination and multiplication.
This method is similar to elimination by addition and subtraction, however one
more step
is added. If the coefficients of one of the variables are not the same, find a
common
multiple for one set and multiply each term by that number. Do the same for the
second
equation. In the end, one set of variables needs to have the same coefficient,
either
positive or negative. Students may want to multiply by a negative so they can
add rather
than subtract the two systems.
As signment :
Glencoe – Algebra 1 chapter 7-4, p.390, #1-12.
DAY 5 – MATRICES
Objectives
Solve systems of equations using matrices.
Students will want a copy of the steps to inputting a matrix system into their
calculator
and how to solve it.
Steps to Using a Matrix to Solve Systems of Equations
*Make sure the variables and numbers are lined up vertically underneath one
another.
2m+3n=9
5m+4n=5
Matrix A is 2X2
Matrix B is 2X1
On the calculator, type 2nd x -1, move over to edit, choose
1 or A. Input the data, ie 2 enter
2 enter 2 enter 3 enter 5 enter 4 enter.
Input matrix 2. Type 2nd x -1, move over to edit, choose 2 or B. Input the data;
2 enter 1
enter, 9 enter, 5 enter.
Go back to home screen, 2nd Mode (Quit).
Type in the inverse of A multiplied by B. 2nd x -1 1 or A, x
-1 * 2nd x -1 2 or B.
The answer is in matrix form. The first variable is on top, m=-3 and the second
variable
is on the bottom, n=5.
Try the following problems from the worksheet “Solving Systems using Matrices”
using
the steps above.
As a final review, students need to be able to solve equations given any format.
A work
sheet is attached (Systems of Equations – look for the tiger) that can be used
in class or as
homework. (No word problems)
ASSESSMENT – See attached quiz
HMWK – Solving Systems by Substitution
Please solve the following systems of equations for x & y using the substitution
method.
Check your work.
1.
y=3x
x+2y=-21
2.
x=4y
2x+3y=44
3.
x+5y=-3
3x-2y=8
4.
c-5d=2
2c+d=4
5.
5r-s=5
-4r+5s=17
