Download system of linear equations

Linear Equations
Systems of Linear Equations - Introduction
Objectives:
• What are Systems of Linear Equations
• Use an Example of a system of linear equations
If we have two linear equations, 𝑦 = 𝑥 + 2 and 𝑦 = 3𝑥 − 6, can these two equations equal
one another?
𝑦 =𝑥+2
𝑦 = 3𝑥 − 6
x
f(x)=x+2
f(x)=3x-6
2
4
0
4
6
6
6
8
12
So what does it mean when two linear
equations equal one another?
The equations must intersect.
Linear Equations
Systems of Linear Equations - Introduction
Objectives:
• What are Systems of Linear Equations
• Use an Example of a system of linear equations
Why do we care?
P
R
I
C
E
Imagine you have a company selling your favorite product.
As the demand of your product
goes down, the quantity of
product in your stores goes up.
Demand
In order to sell your product,
the price must go down.
Supply
As your product supply goes
up, your cost to produce your
product goes up.
Where these two lines
intersect, is the price you
should charge to break even.
QUANTITY
Can you see the inequality
possibilities?
Linear Equations
Systems of Linear Equations - Introduction
Objectives:
• What are Systems of Linear Equations
• Use an Example of a system of linear equations
A farmer has 100 animals consisting of cows, pigs, and chickens. The farmer takes these
animals to market and sells them for $100. If the farmer sells the cows for $10, the pigs for
$3 and sells the chickens 2 for $1, how many of each animal does he sell if there are no
partial animals sold?
What do we know?
Let C = # of cows, let P = # of pigs, and S = # of chickens.
Total number of animals sold: C + P + S = 100
Total cost of the animals sold: 10C + 3P + 0.5S = 100
We have two equations, but three variables.
The number of each animal sold must be greater than one and have no fractions.
Linear Equations
Systems of Linear Equations - Introduction
Objectives:
• What are Systems of Linear Equations
• Use an Example of a system of linear equations
Many times we can solve for one variable and then substitute that expression into a second
equation. There cannot be many cows, so lets solve an equation in terms of S.
C + P + S = 100; then S = 100 – C – P.
Substituting: 10C + 3P + 0.5(100 – C – P) = 100
10C + 3P + 50 – 0.5C – 0.5P = 100
9.5C + 2.5P = 50
19C + 5P = 100
P=-
19
5
C +20; this y = mx + b
If we think about this, the number of cows can not be zero, why?
The number of cows must be a multiple of 5, why?
Therefore, the number of cows must be?
5
Can not be greater than 10, why?
Linear Equations
Systems of Linear Equations - Introduction
Objectives:
• What are Systems of Linear Equations
• Use an Example of a system of linear equations
Knowing one variable in our three variable system of linear equations means we now have
two equations and two variables. We can now solve the other two variables.
C + P + S = 100 or 5 + P + S = 100 or P + S = 95
50 + 3P + 0.5S = 100 or 3P + 0.5S = 50
1) P + S = 95
2) 3P + 0.5S = 50
If we multiply the first equation by 3;
1) 3P + 3S = 285
2) 3P + 0.5S = 50
Remember, these equations must be equal;
therefore, subtracting equation 2 from 1 …
2.5S = 235 so S = 94
This means, the farmer sold 5 cows, 94 chickens, and 1 pig.
Linear Equations
Systems of Linear Equations - Tables
Objectives:
• Solving systems of Linear Equations using tables.
A set of linear equations that has more than one variable is called a system of linear
equations. The single pair of variables that satisfies both equations is their unique solution.
One method to solve a system of linear equations is to make a table of values for each
equation and compare each table for the common solution of both equations.
Lionel is x years old and his sister is y years old. The difference in their ages is 1 year. The
sum of 4 times Lionel's age and his sister’s age is 14 years. Find Lionel’s and his sister’s ages.
We have two equations: x – y = 1 and 4x + y = 14.
x–y=1
4x + y = 14
x
2
3
4
x
1
2
3
y
1
2
3
y
10
6
2
In the second table, the first two pairs does not make sense, because Lionel is older
than his sister. Lionel’s age must be 3 years and his sister’s age must be 2 years.
Linear Equations
Systems of Linear Equations - Tables
Objectives:
• Solving systems of Linear Equations using tables.
How can we use a calculator to solve systems of Linear Equations?
Given the two equations : 8x + y = 38 and x – 4y = 13; find the solution of this system.
TI-84
TI-73
2nd TBLSET / Tblstart = 0 / Tbl =1 / Ind:auto Dep:auto
2nd TBLSET / Tblstart = 0 / Tbl =1 / Ind:auto Dep:auto
Solve each equation for y; input into Y2 and Y3
Solve each equation for y; input into Y2 and Y3
2nd TABLE / arrow down until Y values are equal
2nd TABLE / arrow down until Y values are equal
The unique solution for this system of linear equations is x = 5; y = -2.
Bookwork: Math in Focus – Practice 5.1; page 196, problems 1-12.
Linear Equations
Systems of Linear Equations - Elimination
Objectives:
• Solving systems of Linear Equations using Elimination.
When equations become more complicated, the use of tables may not be the best
method to solve the system.
𝑥+𝑦 =8
Lets look at two equations:
𝑥 + 2𝑦 = 10
Using Bar Models:
x
y
x
y
=8
y
= 10
The difference in length of the two bars is 2 units; therefore, y = 2 because the one ybar must equal the difference of 2 units.
If;
x
Then x must equal 6 units.
2
=8
Linear Equations
Systems of Linear Equations - Elimination
Objectives:
• Solving systems of Linear Equations using Elimination.
What did we do to solve the previous system of linear equations?
We eliminated one variable to solve for the other variable, then substituted the
known variable into one of the equations to solve for the eliminated variable.
𝑥+𝑦 =8
𝑥 + 2𝑦 = 10
If both equations have a variable with the same coefficient, elimination of that
variable by subtraction will yield one equation with only one variable.
𝑥 + 2𝑦 − 𝑥 + 𝑦 = 10 − 8
𝑥 + 2𝑦 − 𝑥 − 𝑦 = 2
𝑥 − 𝑥 + 2𝑦 − 𝑦 = 2
𝑦=2
Substituting y = 2 into the first equation yields x = 6.
The solution is then given by x = 6, y = 2
Linear Equations
Systems of Linear Equations - Elimination
Objectives:
• Solving systems of Linear Equations using Elimination.
We know that addition is the opposite of subtraction. Given the following equations:
4𝑥 + 𝑦 = 9
3𝑥 − 𝑦 = 5
We can eliminate y by adding the two equations:
(4𝑥 + 𝑦) + (3𝑥 − 𝑦) = 9 + 5
4𝑥 + 3𝑥 + 𝑦 − 𝑦 = 14
7𝑥 = 14
𝑥 = 2, 𝑦 = 1
We were able to eliminate a variable by using addition.
This method works well when a variable has the same coefficient in both equations.
Linear Equations
Systems of Linear Equations - Elimination
Objectives:
• Solving systems of Linear Equations using Elimination.
What can we do when there is no common coefficient?
2𝑥 + 3𝑦 = 7
𝑥 + 6𝑦 = 8
Using Bar Models we see:
x
x
x
y
y
y
y
y
y
y
y
=7
y
=8
By using a copy of the second bar model we can redraw the equations:
x
x
y
y
y
x
x
y
y
y
=7
y
y
y
y
y
y
y
y
y
= 16
By using elimination, y = 1. Our solution becomes x = 2, y = 1
We solved this system by multiplying the second equation by 2, giving a common
coefficient for x, then using the elimination method.
Bookwork: Math in Focus – Practice 5.2; page 209, problems 1- 9.
Linear Equations
Systems of Linear Equations - Substitution
Objectives:
• Solving systems of Linear Equations using Substitution.
You have seen how elimination can simplify two equations with two variables into one
equation with one variable. Lets look at our first example again:
𝑥+𝑦 =8
𝑥 + 2𝑦 = 10
x
y
x
y
=8
y
= 10
You can redraw the first bar graph to look like:
=8−
x
y
Then redraw the second bar graph to look like:
8−
y
y
y
= 10
Solving for y; y = 2, then substituting into the first equation yields x = 6.
Linear Equations
Systems of Linear Equations - Substitution
Objectives:
• Solving systems of Linear Equations using Elimination.
By solving for one variable in one equation, you can substitute that value into the second
equation and solve for the one variable. Lets look at a second example algebraically.
3𝑥 − 𝑦 = 18
𝑦 =𝑥 −4
Here, the second equation is already solved for y. If we substitute this value of y into the
first equation…
3𝑥 − 𝑥 − 4 = 18
3𝑥 − 𝑥 + 4 = 18
2𝑥 + 4 = 18
2𝑥 = 14
𝑥 = 7 ;𝑦 = 3
Question: could you solve for x in the second equation and substitute for y in the first
equation?
Bookwork: Math in Focus – Practice 5.2; page 209, problems 10 - 18.
Linear Equations
Systems of Linear Equations - Choose
Objectives:
• Solving systems of Linear Equations using - Choose the method.
You have seen that both methods, elimination and substitution, can be used to solve a
system of linear equations.
Both methods find the same solution set, don’t they? This is because two lines can only
intersect at one point; therefore, there is only one solution.
In Algebra II you will see multiple solutions because Algebra II has functions with
exponents which changes the line into a curve.
Which method is the best method?
That depends on which method you like to use and what the equations look like.
Would you use substitution for the system; 3𝑝 + 2𝑞 = 1 𝑎𝑛𝑑 2𝑝 − 5𝑞 = −12 ?
No, because whatever variable you solved for, there would be fractions.
Bookwork: Math in Focus – Practice 5.2; page 209, problems 19 – 25.
Linear Equations
Inconsistent and Dependent Systems of
Linear Equations
Objectives:
• Understand and identify inconsistent systems of linear equations
• Understand and identify dependent systems of linear equations
You have seen that given two linear equations, the unique solution is one point.
What happens if a system does not have a unique solution?
Given the following system of linear equations:
2𝑥 + 𝑦 = 1 If we multiply the first equation
4𝑥 + 2𝑦 = 4 by 2 and solve by elimination…
4𝑥 + 2𝑦 = 2
4𝑥 + 2𝑦 = 4
We get 0 = 2; this is a false statement; meaning this system does not have a solution.
Lets look at the coefficients of these equations; 2, 1, and 1 and 4, 2, and 4.
Notice the ratios of these coefficients. A and B are the same ratio; however, C is a
different ratio.
Linear Equations
Inconsistent and Dependent Systems of
Linear Equations
Objectives:
• Understand and identify inconsistent systems of linear equations
• Understand and identify dependent systems of linear equations
Lets look at the slopes of these equations: y = -2x + 1 and y = -2x + 2.
The slopes are the same; meaning the lines are parallel.
Parallel lines do not intersect; therefore, there is no unique solution.
A linear system that does not have a unique solution is said to be inconsistent or has no
solution.
Linear Equations
Inconsistent and Dependent Systems of
Linear Equations
Lets consider the following system of linear equations:
𝑥 + 2𝑦 = 2 If we use substitution to solve
𝑥 = 2 − 2𝑦 from equation 1.
2𝑥 + 4𝑦 = 4 this system…
Substituting this into equation 2 yields, 2 2 − 2𝑦 + 4𝑦 = 4
Simplifying yields 4 = 4. This is always true, indicating that all values of x and y are
solutions.
What happens when you graph the two equations? Solving for y in both equations
1
yields 𝑦 = − 𝑥 + 1. These are the same equation resulting in the same line;
2
therefore all points are a solution.
Looking at the coefficients in the two equations; 1, 2, and 2 and 2, 4, and 4, the ratios
of A, B, and C are the same. This means the equations are the same, doesn’t it?
A system of equations that have an infinite number of solutions, or has the same slope
and same y-intercepts, or if one equation is a multiple of another, the system is a
dependent system or has infinite solutions.
Bookwork: Math in Focus; page 234, problems 1 – 14.