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Solving Systems by Graphing
The previous section covered two key topics:
1) Graphing “and” and “or” statements in one variable,
2) Solving inequalities in one variable.
The topic of this section is solving “and” statements in
two variables.
This symbol is read, “and”.
It is used to define a system. A system implies the
equations are both, or all, needed … the equations go
together to describe some situation.
Solving Systems by Graphing
Solutions to systems of equations in two variables are
ordered pairs. They are the points of intersection of the
equations in the coordinate plane.
Consistent systems have at least one solution, at
least one point of intersection. A system of equations
that does not intersect is called an inconsistent
system.
Solving Systems by Graphing
Solve the system below by graphing on paper in a
coordinate plane.
In other words, find the point(s) of
intersection.
How many solutions to the system?
Find the solutions and test them in
the equations.
(-1,-1)(3,7)
Solving Systems by Graphing
The book lists solutions to systems three different
ways. They are:
(-1,-1)(3,7)
{(-1,-1)(3,7)}
1y
1x
7y
3xor
7
3 or
1
1
With matrices in the last chapter, there’s a fourth way
Solving Systems by Graphing
One of the weaknesses of solving systems by graphing
is that the method sometimes produces, at best,
estimates.
Graph
Type the equations into your calculator like you are
graphing and estimate the solution to the system by
tracing or using tables.
(.125,.625)
Solving Systems by Graphing
A fence will enclose a rectangular piece of land along a
river. Because the river is on one side of the rectangle,
that side of the fence is not needed. There is 80 m of
fencing material and the fence will enclose 500 m2.
Find the dimensions of the region.
500ws
802sw Graph on your
calculator.
Solving Systems by Graphing
A fence will enclose a rectangular piece of land along a
river. Because the river is on one side of the rectangle,
that side of the fence is not needed. There is 80 m of
fencing material and the fence will enclose 500 m2.
Find the dimensions of the region.
s
500w
2s-80w Use tables to find the
solutions to the nearest
integer.
(8,65)(32,15)
Solving Systems with Substitution
Solve with substitution.
We’ve seen it’s sometimes difficult or impossible to
find exact solutions to systems by graphing. Today we
begin work on methods that result in exact solutions.
3(-2) – 4y = -9
-6 – 4y = -9
-4y = -3
y = 0.75
You’re not finished. Recall,
solutions to systems in two
variables are ordered pairs.
(-2, 0.75)
Solving Systems with Substitution
Solve with substitution.
Which equation would be easiest to solve for one of
the variables?
y = -3x + 9
Substitution now requires that you plug this into the
other equation for y.
5x + 4(-3x + 9) = 8 Solve for x.
x = 4
(4,-3). Check.
Solving Systems with Substitution
Solve with substitution.
Which equation would be easiest to solve for one of
the variables?
y = 2x + 9
Substitution now requires that you plug this into the
other equation for y.
3x – 4(2x + 9) = -11 Solve for x.
x = -5
(-5,-1). Check.
Solving Systems with Substitution
Solve with substitution.
x + 2x + x + 3 = -5
x = -2
(-2,-4,1). Check.
Solving Systems with Substitution
Solve with substitution.
x (2x) = 4
2x2 = 4
x2 = 2
2x
22 ,222 ,2
Solving Systems with Substitution
Blue paint is mixed with three times as much red paint
to get 2 gallons of violet paint. How much blue, and
how much red paint was used?
3b + b = 2
4b = 2
b = 0.5
0.5 gal of blue paint,
1.5 gal of red paint
Solving Systems with Linear Combination
The system below can be solved easily with
substitution, but there’s probably a faster method.
If you simply add the equations, y is
gone. That was the goal of substitution
too. Eliminate one variable so you can
solve for the other.
x + y = 9
2x – y = 2
3x = 11
x = 11/3
(11/3,16/3)
Solving Systems with Linear Combination
Solve
If you add them, neither the x or y disappear. The
same is true if you subtract. Multiply both sides of
the bottom equation by 2 and subtract. In other
words, multiply both sides of the bottom equation
by -2 and add.
2x + 4y = 3
-2x + 20y = 6
24y = 9
y = 9/24 = 3/8
(3/4,3/8). Check
with linear combination.
Solving Systems with Linear Combination
Solve
Sometimes, both sides of BOTH equations need to be
multiplied by some number in order to eliminate a
variable. That is the case here. What are our options?
6x + 10y = 14
-6x + 12y = -36
22y = -22
y = -1
(4,-1). Check
with linear combination.
Top times 2, bottom times -3. This gets rid of x. Or top
times 4, bottom times 5. This gets rid of y.
Solving Systems with Linear Combination
Solve
-6x – 15y = -9
6x + 15y = 9
0 = 0
BOTH x and y disappeared. Now what? These
equations represent the same line. Here’s how to
write the solution set:
{(x,y)| 2x + 5y = 3} This says, “All ordered pairs on the
line 2x + 5y = 3 are solutions.”
with linear combination.
Solving Systems with Linear Combination
Solve
-6x – 15y = -12
6x + 15y = 9
0 = -3
These equations represent parallel lines. They have
no solution, no intersection. The solution set is Ø.
with linear combination.
Solving Systems with Linear Combination
In some applications, a table of information is
supplied. If one isn’t, it sometimes helps to create a
table. The activity on page 322 has a table. Examine
it and make sure you understand where the system
of equations comes from.
You may skip example 2 p.323. It’s possible to solve
systems of three equations in three variables with
linear combination, but the next method we learn is
much more convenient. You will notice problem 17
has three equations, but since we know z, it easier.
Solving Systems with Linear Combination
Two large and one medium pizza cost $46.35. One
large and one medium pizza cost $29.40. Find the
cost of a medium pizza.
2l + m = 46.35
-l – m = -29.40
l = 16.95
A medium pizza costs $12.45.
The Inverse of a 2 x 2 Matrix
What is the multiplicative inverse of ¾?
4/3. Another word for multiplicative inverse is
reciprocal. It’s the number you’d multiply by ¾ to get 1.
What do you think the multiplicative inverse for a matrix
would mean?
It’s the matrix you’d multiply by another to get the
identity matrix.
dc
ba
10
01
dc
ba
qp
nm
qp
nm
If the inverse of is ,
10
01
qp
nm
dc
baand
The Inverse of a 2 x 2 Matrix
To find the inverse of the 2 x 2 matrix
dc
ba
dc
ba
43
1-2
something called the determinant. The determinant of
, we use
is ad – bc. Find the determinant of
The determinant is 11.
I’ll call the matrix
dc
bamatrix M. The determinant
is used to find the inverse, denoted M-1. Here’s how…
The Inverse of a 2 x 2 Matrix
dc
baM = M-1 =
Scalar multiplication by , then
Switch a and d around.
Then b and c stay, but are opposites.
The Inverse of a 2 x 2 Matrix
dc
baM = M-1 =
Find the inverse of
43
1-2The determinant is 11.
M-1 =
To test, multiply.
The Inverse of a 2 x 2 Matrix
dc
baM = M-1 =
Find the inverse of
M-1 = Check by multiplying.
The Inverse of a 2 x 2 Matrix
dc
baM = M-1 =
Find the inverse of
None exists. If the determinant is zero, there is
no inverse.
The activity on p.332 explains how to use matrices to
encode and decode a message. This is how it works.
Here’s a message to encode and decode: GO_HOME I’ll use the correspondence key below. A B C D E F G …
1 2 3 4 5 6 7 …
7 15 27 8 15 13 5
Store these in a 2 x 2 matrix: Choose a coding matrix. I’ll use this matrix:
Multiply it by the message matrix.
=
Give this and the coding matrix to a partner. They’ll
multiply it by the inverse of the coding matrix to get the
original message. Decode it.
Solving Systems with Matrices
Solve with matrices.
The system must be written in matrix form. A B
Just like solving 3x = 9, where
you’d divide both sides by 3, or
multiply both sides by 1/3, we’ll
multiply both sides by A-1. A-1 A A-1 B
Solving Systems with Matrices
Solving Systems with Matrices
If you don’t need to show work and you’d like to get
the answer quickly, store A and B in your calculator
and ask it to calculate A-1 · B.
A B
Solving Systems with Matrices
Solve
A system of three
equations in three
variables is most easily
solved with matrices. A B
Store A and B in your calculator and calculate A-1 · B.
(4,-2,3)
Solving Systems with Matrices
We skipped problem 9 on page 326. Write a system
for the problem and solve it with matrices.
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