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Unit 5 – Systems of Linear Equations and Inequalities

This unit begins by ensuring that students understand that solutions to equations are points that make the equation true, while solutions to systems make all equations (or inequalities) true.  Graphical and substitution methods for solving systems are reviewed before the development of the Elimination Method.  Modeling with systems of equations and inequalities is stressed.  Finally, we develop the idea of using graphs to help solve equations.

Solutions to Systems and Solving by Graphing

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Solving Systems by Substitution

Properties of Systems and Their Solutions

The Method of Elimination

Modeling with Systems of Equations

Solving Equations Graphically

Solving Systems of Inequalities

Modeling with Systems of Inequalities

Unit Review

Unit #5 Review – Systems of Linear Equations and Inequalities

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit #5 Assessment Form A

EDITABLE ASSESSMENT

Unit #5 Assessment Form B

Unit #5 Assessment Form C

Unit #5 Assessment.Form D

Unit #5 Exit Tickets

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form A

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form B

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form C

U05.AO.01 – Solving Equations Graphically – Extra Practice (After Lesson #6)

EDITABLE RESOURCE

U05.AO.02 – Additional Modeling with Linear Systems

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Systems of Equations (Word Problems)

These lessons, with videos, examples and step-by-step solutions help Grade 8 students learn how to analyze and solve pairs of simultaneous linear equations.

Related Pages Systems of Equations - Graphical Method Solving Equations Common Core for Grade 8 Common Core for Mathematics More Math Lessons for Grade 8

A. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

B. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6 .

C. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Common Core: 8.EE.8c

Suggested Learning Targets

• I can identify the solution(s) to a system of two linear equations in two variables as the point(s) of intersection of their graphs.
• I can describe the point(s) of intersection between two lines as the points that satisfy both equations simultaneously.
• I can define “inspection.”
• I can solve a system of two equations (linear) in two unknowns algebraically.
• I can identify cases in which a system of two equations in two unknowns has no solution.
• I can identify cases in which a system of two equations in two unknowns has an infinite number of solutions.
• I can solve simple cases of systems of two linear equations in two variables by inspection.
• I can estimate the point(s) of intersection for a system of two equations in two unknowns by graphing the equations.
• I can represent real-world and mathematical problems leading to two linear equations in two variables.

Systems of equations word problem (coins)

Example: A man has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is $2.75, how many dimes and how many quarters does he have?

Word problem using system of equations (investment-interest)

Example: A woman invests a total of $20,000 in two accounts, one paying 5% and another paying 8% simple interest per year. Her annual interest is $1,180. How much did she invest in each rate?

Systems of Equations Word Problems

Example: The sum of two numbers is 16. One number is 4 less than 3 times the other. Find the numbers.

Systems of Equations (word problems)

Example: Two times a number plus ten times a second number is twenty. Thirty times the second number plus three times the first number is 45. What are the two numbers?

Systems of Equations-Word Problems

How to solve a word problem involving a system of 2 equations with 2 variables?

Example: Three coffees and two muffins cost a total of 7 dollars. Two coffees and four muffins cost a total of 8 dollars. What is the individual price for a single coffee and a single muffin?

How to translate words or word problems into a systems of equations?

Example: A coin collection is made up of 34 coins comprised of nickels and dimes. The total value of the collection is $1.90. How many dimes and nickels made up this collections?

Systems of Equations - word problems Examples of setting up word (or application) problems solved by a system of equations.

Example: For some reason, in our math class, there are 14 more boys than there are girls. If a total of 32 students are in the class, how many boys and how many girls are there?

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System-of-Equations Word Problems

Exercises More Exercises

Many problems lend themselves to being solved with systems of linear equations. In "real life", these problems can be incredibly complex. This is one reason why linear algebra (the study of linear systems and related concepts) is its own branch of mathematics.

In your studies, however, you will generally be faced with much simpler problems. What follows are some typical examples.

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?

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System of Equations Word Problems

In the past , I would have set this up by picking a variable for one of the groups (say, " c " for "children") and then use "(total) less (what I've already accounted for)" (in this case, " 2200 –  c ") for the other group. Using a system of equations, however, allows me to use two different variables for the two different unknowns.

number of adults: a

number of children: c

With these variables, I can create equations for the totals they've given me:

total number: a + c = 2200

total income: 4 a + 1.5 c = 5050

Now I can solve the system for the number of adults and the number of children. I will solve the first equation for one of the variables, and then substitute the result into the other equation:

a = 2200 – c

4(2200 – c ) + 1.5 c = 5050

8800 – 4 c + 1.5 c = 5050

8800 – 2.5 c = 5050

–2.5 c = –3750

Now I can back-solve for the value of the other variable:

a = 2200 – (1500) = 700

I have values for my two variables. I can look back at my definitions for the variables to interpret these values. To answer the original question, there were:

1500 children and 700 adults.

You will probably start out with problems which, like the one above, seem very familiar. But you will then move on to more complicated problems.

The sum of the digits of a two-digit number is 7 . When the digits are reversed, the number is increased by 27 . Find the number.

The trick here is to work with the digits explicitly. I'll use " t " for the "tens" digit of the original number and " u " for the "units" (or "ones") digit. I then have:

The ten's digit stands for "ten times of this digit's value". Just as "26" is "10 times 2, plus 6 times 1", so also the two-digit number they've given me will be ten times the "tens" digit, plus one times the "units" digit. In other words:

original number: 10 t + 1 u

The new number has the values of the digits (represented by the variables) in reverse order. This gives me:

new number: 10 u + 1 t

And this new number is twenty-seven more than the original number. The keyword "is" means "equals", so I get:

(new number) is (old number) increased by (twenty-seven)

10 u + 1 t = (10 t + 1 u ) + 27

Now I have a system of equations that I can solve:

10 u + t = 10 t + u + 27

First I'll simplify the second equation:

9 u – 9 t = 27

u – t = 3

After reordering the variables in the first equation, I now have:

Adding down , I get:

Then t = 2 . Back-solving, this means that the original number was 25 and the new number (gotten by switching the digits) is 52 . Since 52 – 25 = 27 , this solution checks out.

The number is 25 .

Find the equation of the parabola that passes through the points (–1, 9) , (1, 5) , and (2, 12) .

Recalling that a parabola has a quadratic as its equation, I know that I am looking for an equation of the form ax 2  +  bx  +  c = y . Also, I know that points are of the form ( x ,  y ) . Practically speaking, this mean that, in each of these points, they have given me values for x and y that make the quadratic equation true. Plugging the three points in the general equation for a quadratic, I get a system of three equations, where the variables stand for the unknown coefficients of that quadratic:

a (–1) 2 + b (–1) + c = 9

a (1) 2 + b (1) + c = 5

a (2) 2 + b (2) + c = 12

Simplifying the three equations, I get:

1 a – b + c = 9

1 a + b + c = 5

4 a + 2 b + c = 12

I won't display the solving of this problem, but the result is that a  = 3, b  = –2, and c  = 4 , so the equation they're wanting is:

y = 3 x 2 – 2 x + 4

You may also see similar exercises referring to circles, using:

x 2 + y 2 + bx + cy + d = 0

...or other conics, though parabolas are the most common. Keep in mind that projectile problems (like shooting an arrow up in the air or dropping a penny from the roof of a tall building) are also parabola problems, using:

–( 1 / 2 ) gt 2 + v 0 t + h 0 = s

...where h 0 is the original height, v 0 is the initial velocity, s is the height at time t , usually measured in seconds, and g refers to gravity, being 9.8 if you're working in meters and 32 if you're working in feet).

All of these different permutations of the above example work the same way: Take the general equation for the curve, plug in the given points, and solve the resulting system of equations for the values of the coefficients. Warning: If you see an exercise of this sort in the homework, be advised that you may be expected to know the forms of the general equations (such as " ax 2 + bx + c = y " for parabolas) on the next text.

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Solving Systems of Equations Word Problems Worksheets

We have previously learned about systems of equations. This is when you have multiple linear equations that have the same value for their unknown variables. this allows us to manipulate these equations in many different ways to determine those unknown variables. These systems are evident in many different real-world ways. Students will work with systems of equations that can be found within word problems in this problem set. The worksheets and lessons will show students how to setup up these problems and how to solve them.

Aligned Standard: HSA-REI.C.5

• Carpenter's Project Step-by-step Lesson - We're making chairs, but at what rate and how often?
• Guided Lesson - Making muffins, pearl necklaces, and making sandwiches.
• Guided Lesson Explanation - Create the system first and then find the sweet spot that they both fit into.
• Practice Worksheet - Our scenarios include candle making, buying books and fruit juices.
• Matching Worksheet - Fill in the missing numbers to make each word problem mathematically correct.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Some of these problems were so hard to get right that it took several tries.

• Homework 1 - A test has 20 questions worth a total of 100 points. There are two question types on the test. Matching questions are worth 4 points and multiple choice questions are worth 6 points. How many multiple choice and matching questions are there each?
• Homework 2 - 3 small vases and one large vase hold 6 liters of water. If you poured a full large vase into a small vase and filled it, you would still have 2 liters of water left in the large vase. How much water can a large vase hold?
• Homework 3 - Billy went to the fun park. He rode the water slide 5 times and the Ferris wheel twice. He spent $20 on the rides. Jenny rode the water slide 3 times and the Ferris wheel 3 times. Jenny spent $21 on rides. How much does it cost to go on the water slide?

Practice Worksheets

The flow here is set up to make students work a little harder as they go along.

• Practice 1 - Shelly wants to make cookies. She has two different sized cooking pans. She only uses one pan per flavor of cookie. On Saturday, she makes 12 pans of chocolate cookies and 10 pans of vanilla cookies for a total of 312 cookies. On Sunday, She makes 5 pans of chocolate cookies and 3 pans of vanilla cookies for a total of 116 cookies.
• Practice 2 - Ash sells dumplings at the local farmer’s market. On Monday, he sold 12 veggie dumplings and 15 meat dumplings for $69. On Tuesday, he sold 18 veggie dumplings and 7 meat dumplings for $57.
• Practice 3 - The sum of two numbers is 26. The difference of the same two numbers is 8. What are the numbers?

Math Skill Quizzes

The quizzes were hard to get set correctly. It took me a while to get the numbers to balance.

• Quiz 1 - Kelly had a great basketball game today. She made 39 points with 16 different shots. If she only made two and three point shots, how many of each did she make?
• Quiz 2 - At the movies, an adult ticket costs $9 and a kid’s ticket costs $6. If Mr. Fluffy brought 9 people and spent $69, how many kids did he pay for?
• Quiz 3 - A Snickers bar has 150 more calories than a large chocolate chip cookie. Donnie has 3 Snickers bars and 4 chocolate chip cookies for a total of 1,150 calories. How many calories are in each Snickers bar and Chocolate chip cookie?

What Are Systems of Equations in Math?

When you consider systems in mathematics, this normally refers to a series of linear equations that share the same values for their variables. Thus, we must learn the concepts of systems when we are learning algebra. Systems of the equation consist of at least two equations that have the same variables in them. In the concept, the number of equations is directly proportional to the number of variables present in them. Let us take an example which consists of variables x and y. You can communicate these types of problems in algebraic or narrative form.

There are number of different methods you can use to find the values of the variables within these systems. Here is an example of a system of equations: x + y = 6 and -3x + y = 2. Now, we need to solve them simultaneously, to get the values of x and y. When you solve them you will come across a condition. Let us resolve them. x = 6 - y : eq (a)| Here, we will place the value of x in the second equation. -3(6-y) + y = 2 unto -18 + 3y + y = 2 unto 4y = 18 + 2 unto y = 5. Now, we will place the value of y in eq (a). x = 6 - 5 therefore x = 1. The solution set is (1, 5). The condition is that on the graph, both lines will intersect at some point. Your goal by substituting values into the other equation is to locate this point. The best thing about this problem is that it easy to double check your solution. All you need to do is to plug the x and y value of your solution into both equations. If you are correct, they will work for both equations. For the purposes of this topic, we will focus on the narrative form with this topic, but to see it written algebraically is good to get us going.

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Homework 6 System Word Problems

Homework 6 System Word Problems - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Systems of equations word problems, Word problem practice workbook, Systems word problems, Proportions word problems, Name date per, 1 read carefully and underline key words write a let, Martha ruttle, Lesson 11 measurement and units of measure.

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1. Systems of Equations Word Problems

2. word problem practice workbook, 3. systems word problems, 4. proportions word problems, 5. name date per, 6. 1 : read carefully and underline key words : write a let ..., 7. martha ruttle, 8. lesson 11: measurement and units of measure.

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Homework 6 Systems Word Problems

Displaying top 8 worksheets found for - Homework 6 Systems Word Problems .

Some of the worksheets for this concept are Unit 6 quadratic word problems, Systems word problems, Systems of linear equations in two variables mat 1190, Systems of equations word problems, Chapter 6 resource masters, Word problem practice workbook, Name date per, College level mathematics pretest.

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• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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• Middle School Math Solutions – Equation Calculator Welcome to our new "Getting Started" math solutions series. Over the next few weeks, we'll be showing how Symbolab...

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