Hartfield – Intermediate Algebra (Version 2014-2D) …faculty.ung.edu/.../courses/2014-3-fall/0099_notes_unit2.pdfHartfield – Intermediate Algebra (Version 2014-2D) Unit 2 | Page

Hartfield – Intermediate Algebra (Version 2014-2D) Unit 2 | Page 1

Topic 2‐1 Systems of Equations

A system of equations is a collection of two or more equations with the same solution set. The equations are simultaneously satisfied. A solution to a system of equations should contain values for each variable that solve each equation in the system. We will be focusing on systems of two linear equations of two variables. Henceforth, our solutions will be ordered pairs.

Are the ordered pairs (3, 1) and (1, 2) solutions to the following system of equations?

2 52 3 3x yx y

We will cover 3 ways of solving a system of equations.

1. finding the intersection point on the graphs of the equation

2. applying the substitution method 3. applying the elimination method

(also known as the addition method)


Solve the system of equations by finding the point of intersection, if one exists.

5

2 4y xy x


1

3 5y x

x y



3 2 52 8x yx y

Consistency ‐ property of having a solution Independence ‐ property of having uniqueness

Most systems of linear equations are consistent and independent. That is, they have a solution and are not a collection of the same line. With a consistent and independent system there is exactly one solution which occurs at an intersection point.

Some systems of linear equations are inconsistent. That is, the two equations never intersect and thus have no solution. Graphically, these lines must be parallel and unique.

Some systems of linear equations are dependent. That is, the two equations are really one equation producing one line. These systems have infinitely‐many solutions along the common line. Graphically, these lines are coincident.


Solve the system of equations by finding the point of intersection, if one exists. If the system has infinitely‐many solutions or it has no solutions, state as such.

Solve the system of equations by finding the point of intersection, if one exists. If the system has infinitely‐many solutions or it has no solutions, state as such.

3 16 2 2

y xx y

6 3 122 5

x yy x


Topic 2‐2 Solving Systems: Substitution Method

With the substitution and elimination methods, the goal will be to take systems with multiple equations and multiple variables and merge them into a single equation involving a single variable.

12 3

y xy x

Substitution Method for Solving Systems of Equations 1. Solve either equation for x or y.

2. Plug that equation in for the variable in the other equation.

3. Solve the created equation involving one variable.

4. Plug the solution found in step 3 into any equation involving two variables & solve.

5. Write your solution as an ordered pair.

2 3 52 6

x yx y


Solve and classify the system of equations.

2 12 7x yx y


4 2 22 3 19x yx y



6 5 133 2 11x yx y

Solving systems of linear equations by the substitution method or the elimination method works obviously when the system is consistent and independent. To recognize when the system is inconsistent (having no solutions) or dependent (having infinitely‐many solutions along a common line), look for systems which do not produce unique solutions:

An inconsistent system should produce a contradiction when the equations are merged. A dependent system should produce an identity statement when the equations are merged.


Solve and classify the system of equations. If the system has no solutions or infinitely‐many solutions, state that.

3 24 12 8

x yx y


4 2 162 6x yx y


Topic 2‐3 Solving Systems: Elimination Method

Elimination Method for Solving Systems of Equations 1. As necessary, multiply one or both coefficients by a positive or negative whole number so that a column of coefficients have the same absolute value but different signs.

2. Add the equations to eliminate a variable and create a new equation.

3. Plug the solution found in step 2 into any equation involving two variables & solve.

4. Write your solution as an ordered pair.


3 114 2 2

x yx y

2 75

x yx y



3 172 11

x yx y


2 3 75 2 8x yx y



5 2 233 5 29x yx y


42 8

63

x y

x y



3 2 106 4 16x yx y


4 2 82 4x yx y


Topic 2‐4 Solving Applications of Systems

A very important use of systems of equations is in solving real‐world scenarios. When the values of two things are unknown, the only way to get a definitive solution to a situation is by having two equations that provide information about a system. Your responsibility will be to create systems of equations out of statements and then appropriately solve them and verbally state the solution.

Solving Applied Systems Problems

1. Read the question carefully.

2. Make notes, charts, or images—as necessary—to better understand the question.

3. Translate the unknowns—as appropriate and necessary—

into mathematical variables. Clearly define your variables with words.

4. Translate the problem into mathematical statements. If

your variables are appropriately defined, your equations should be readable as complete sentences.

5. Solve

6. Check your solutions and interpret the solutions

appropriately using the definitions of the variables and the mathematical statements you form.


Part a: Basic applied systems In my pocket are fourteen coins, all of them quarters or dimes. If the total amount of change is $2.75, how many quarters and how many dimes do I have?

The Gold Star Baseball Team sold 311 tickets for an afternoon game. Regular admission tickets cost $6 while youth tickets cost $2. If the total receipts were $1542, how many tickets of each type were sold?


In 2000, the top two destinations for conventions were Washington, D.C. and Orlando, Florida, which between them, hosted 353 conventions. If Washington D.C. had 27 more conventions than Orlando, find the number of conventions held in each location.

A company rents trucks to help groups move their equipment to and from a presentation site. On two occasions the manager rented a truck at the same daily rental fee and the same cost per mile. On the first date the manager drove 50 miles and the bill was $75. On the second date the manager drove 200 miles and the bill was $120. How much was the daily rental fee and what was the cost per mile?


Part b: Systems involving blends and rates

Blends ‐ you have so much of one object and so much of a second object, how much will you have when you mix it together? Remember: simple interest = principle x rate x time distance = rate x time Many problems involving interest are a rate and a blend at the same time.

A Candy Barrell shop manager mixes M&M's worth $10 per pound with trail mix worth $6 per pound. Find how many pounds of each she should use to get 40 pounds of party mix worth $7.50 per pound.


A store is selling a mix of wildflower seeds. In the mix is a purple flower that costs $3 per ounce and a blue flower that costs $7 per ounce. If the store wants to create a 12 ounce mix that will sell for $4 per ounce, how much of each type of flower seed should be included in the mix?

A pharmacist needs 500 milliliters of a 20% saline solution but only had 5% and 25% saline solutions available. Find how many milliliters of each he should mix to get the desired solution.


The pharmacist is knocked out by an evil chemist who has decided to replace all the bottles of saline solution with hydrochloric acid solutions. Two bottles of solution contain 3% HCl acid in solution and 15% HCl acid in solution. How much of each solution should be mixed together to make 1.5 L of 7% hydrochloric acid solution.

The company Juiced‐Up wants to market a drink which is 10% real fruit juice. They receive a base containing 4% fruit juice and mix it with pure fruit juice. How much of the base and how much of the pure fruit juice should be mixed together for each 64‐ounce bottle?


Sam Abney invested $9000 one year ago. Part of the money was invested at 6%, the rest at 10%. If the total interest earned in one year was $652.80, find how much was invested at each rate.

Two cyclists start at the same point and travel in opposite directions. One travels 4 mph faster than the other. In four hours they are 112 miles apart. Find how fast each is traveling.


Topic 2‐5 Linear Inequalities & Interval Notation

Equations and inequalities are both statements. Both involve the comparison of two things.

Examples of inequalities:

2 1

5, 4 12, 3 6, 35x

x x x

In an inequality, the comparison implies they cannot simply be equal to each other. Solutions to an inequality, similar to solutions to an equation, are values of the variable which make the statement true.

5x is an example of an inequality which is its own solution statement. A solution inequality is an inequality statement where the variable is by itself on one side.

In most ways you can solve a linear inequality like you solve an equation using properties of addition and multiplication.

Solve each inequality.

3 6x

4 12x


Some inequality statements may take more work to solve. Multiple steps or simplification of one side may be necessary.

Solve.

2 13

5x

Solve.

4 2 3 5 4 2x x x


The one special issue with solving inequalities is that if you divide or multiply by a negative number, you must reverse the inequality direction.

Solve.

4 28x

Solve.

52x

5 20x


We like to graph solution inequalities so we can better see what numbers can be solutions to the original inequality.

When you graph a solution inequality on a number line, only mark where the important number(s) are that establish a “boundary” that separates the solutions from the non‐solutions.

To graph a solution inequality, mark the boundary value, determine which way to shade, and then use a symbol to indicate if the boundary is a solution. If it is, either use a closed dot or a bracket. If it isn’t, either use an open dot or a parenthesis.

Graph.

x < 5

Graph.

x > 9

x < 3

x > ‐7

x < ‐10


Interval Notation Another way to represent the solutions is to write an interval in interval notation.

Interval notation is created by deciding if the solution set has lower and/or upper limits.

a: If a lower limit exists, find a boundary value on the left side of the interval. If a lower limit doesn’t exist, write − instead.

b: If an upper limit exists, find a boundary value on the right side of the interval. If an upper limit doesn’t exist, write instead.

c: If a boundary value is in the interval, put a bracket next to it. If a boundary value is not in the interval, put a parenthesis next to it. If a positive or negative infinity symbol is used, always put a parenthesis next to it.

Solve. Express solutions graphically and in interval notation.

54

2x

x


Topic 2‐6 More With Inequalities

Solve the inequality. Let the unknown number be x.

Five less than a number is greater than or equal to three times the sum of a number and one.

Compound inequalities Some inequality statements are written with three parts using two inequality symbols. These are compound inequalities. In a compound inequality, the inequality symbols must always point the same direction and it is highly preferable that they point to the left (less than/less than or equal to symbols). If the only place a variable is at is in the middle of the compound inequality, it is possible to solve the inequality by applying similar steps to all three parts of the inequality.


Solve. Express solutions graphically and in interval notation. 4 < 2x – 10 < 26

Solve. Express solutions graphically and in interval notation. – 3 < –x + 4 < 1


Applications of Inequalities Uncle Bob's Barbecue caters corporate events. Each plate is $6 and there is an additional charge of $250 for set‐up and delivery for an event. ZXY Communications is hosting a picnic for its employees and families. $1780 has been budgeted for the food at the picnic. How many people can ZXY feed within their budget?

Harvey is moving some large equipment across town.

He will need to rent a moving truck to get the equipment moved. Two local companies will rent a truck for a day:

Store A charges $50 plus 25 cents per mile in travel. Store B charges $30 plus 60 cents per mile in travel.

How many miles could Harvey drive so that Store A is cheaper than Store B? Note: In the real world, prices are usually charged by the whole mile.

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Hartfield – Intermediate Algebra (Version 2014-2D) …faculty.ung.edu/.../courses/2014-3-fall/0099_notes_unit2.pdfHartfield – Intermediate Algebra (Version 2014-2D) Unit 2 | Page