# Math 90 Chapter 7 Lecture Notes

**Section 7.2: Systems of Equations and Substitution
**

The method of substitution for solving systems of linear equations works for any

linear system , but works best if one of the equations in the system can easily be

solved for one of the variables. What we look for in particular is a variable in

either equation that has a coefficient of 1. This makes the substitution method very

easy to use. Work through the following examples to see how easy this method is

to use. Remember that we are trying to find a point where the two lines intersect,

and a point has both an x and a y coordinate .

A. Some sample problems: Solve using the method of Substitution.

1. 2x + y = 1

x – 5y = 17

2. 2x – 4y = -4

x + 2y = 8

3. 7x + 6y = -9

y = -2x + 1

B. Remember that not all systems of linear equations have
one unique solution.

What are the other cases?

1. If the lines are parallel, how many solutions will you find? ______

2. If the lines coincide, how many solutions will you find? _______ .

C. Some rules for solving systems of linear equations

5. If possible, remove by factoring, any constant multiplier that may be in

the equations. This action will not change the equality but will make the

equation simpler.

6. Re-arrange the equations such that both are in the same form. For

instance place all terms on the left with zero on the right.

7. If the two equations are identical, then these equations lie one on top of

the other and there are an infinite number of solutions .

8. If the variable terms are identical but the constants different then these

are parallel lines and there is no solution.

9. If options 3 and 4 are both false then there is one solution.

D. Work through the following examples to see what happens when you encounter

one of the situations described above:

1. 2x + 6y = -18 x + 3y = -9 |
How many solutions? _____________ |

2. 10x + 2y = -6 y = -5x |
How many solutions? _____________ |

E. Some examples of application problems:

1. Find two numbers such that the sum is 76 and the difference

is 12.

2. Two angles are supplementary. One angle is 8 degrees less than

three times the other . Find the measure of each angle.

3. The state of Wyoming is a rectangle with a perimeter of 1280

miles. The width is 90 miles less than the length. Find the length

and the width.

**Section 7.3: Systems of Equations and Elimination**

In this section we will learn another method for solving systems of equations
called

the Elimination Method. In this method, you want to ultimately add the two

equations together and eliminate one of the variables. Work through the
following

examples to see how this is done.

A. The Rational behind this method: Note the following:

If A = B and

C = D then it is true that: A+C = B+D

Because of this fact, we can always add two equations together to get another

true equation. We will use this fact when we use the elimination method for

solving systems of equations.

B. Some sample problems: Solve the following using the Elimination Method. But

before solving these equations let us examine them to determine what kind of

solution we can expect.

1. x – y = 1

-x – y = -7

2. x – y = 4

2x + y = 8

3. –x + 10y = 1

-5x + 15y = -9

4. 2x + 9y = 2

5x + 3y = -8

C. Once again remember that not all systems of equations have a unique solution.

Follow the next two examples to see what happens when we get no solution, or

an infinite number of solutions.

1. 8x – 2y = 2 4x – y = 2 |
How many solutions does this system have? |

How many solutions does this system have? |

**Section 7.4: More Applications using Systems
**

In this section we will practice solving application problems using systems of

equations. The most important part of this process is setting up the system that

models the situation. You can use either the substitution method or the elimination

method to solve the system that you set up.

A. Sample application problems:

1. In winning the 2000 conference finals, the Los Angeles Lakers

scored 69 of their points on a combination of 31 two and three

pointers. How many of each type of shot did they make?

2. Filmworks charges $1.75 for a role of 24-exposure film and $2.25

for a 36-exposure roll of film. Stu bought 19 rolls of film for a

total of $39.25. How many rolls of each type did he buy?

3. From November 2 through January 3, the Bronx Zoo
charges $6.00

for adults and $3.00 for children and seniors. One December day,

a total of $1554 was collected from 394 admissions. How many

adult and how many children/senior admissions were there?

4. Sunflower seed is worth $1.00 per pound and rolled oats
are

worth $1.35 per pound. How much of each would you use to make

50 pounds of a mixture worth $1.14 per pound.

5. Clear Shine window cleaner is 12% alcohol and Sunstream

window cleaner is 30% alcohol. How many ounces of each should

be used to make 90 ounces of a cleaner that is 20% alcohol?

**Section 7.5: Linear Inequalities in Two Variables**

In this section we will learn how to find solutions of linear inequalities in
two

variables.

A. An example of an inequality in one variable. Solve and graph the

solution set for the following inequality:

3(x + 5) ≤ 22

Note: There are an infinite number of solutions to this inequality. Because of
this

we need to describe our solutions and/or graph them as it would be impossible to

list them all.

B. Find and graph as many ordered pair solutions to the following

inequality.

C. What do you notice about the boundary for the solutions
to this

inequality in two variables?

1. _________________________________________________

2. _________________________________________________

D. Graph the solution set for the following inequalities
in two variables:

1. y ≤ 3x + 2

2. 5x + 4y ≥ 20

Remember that when you are graphing the solution set to an
inequality in two

variables that you must first graph the “Boundary Line”. The boundary line will
be

solid when: ______________________ , and the boundary line will be dashed or

dotted when: _______________________ .

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