Solving Systems by Graphing - wizardsdoor.com 5-2to6... · Solving Systems by Graphing One of the weaknesses of solving systems by graphing is that the method sometimes produces,

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.


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.


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)


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


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)


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.


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)



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.



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.



x + 2x + x + 3 = -5

x = -2

(-2,-4,1). Check.



x (2x) = 4

2x2 = 4

x2 = 2

2x

22 ,222 ,2


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)


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.


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


Top times 2, bottom times -3. This gets rid of x. Or top

times 4, bottom times 5. This gets rid of y.


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.”



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.



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.


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


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…


dc

baM = M-1 =

Scalar multiplication by , then

Switch a and d around.

Then b and c stay, but are opposites.


dc

baM = M-1 =

Find the inverse of

43

1-2The determinant is 11.

M-1 =

To test, multiply.


dc

baM = M-1 =

Find the inverse of

M-1 = Check by multiplying.


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



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


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)


We skipped problem 9 on page 326. Write a system

for the problem and solve it with matrices.

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Solving Systems by Graphing - wizardsdoor.com 5-2to6... · Solving Systems by Graphing One of the weaknesses of solving systems by graphing is that the method sometimes produces,