Back To Course

ELM: CSU Math Study Guide15 chapters | 136 lessons

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Did you ever encounter a problem where you seem to have lots of information, but still have a couple of critical numbers missing? In this lesson, we'll practice solving word problems that require us to set up systems of equations.

Lots of things have systems. For example, there's a knitter's system of organizing yarn by color or a chef's system of chopping and organizing ingredients before cooking. Even when society breaks down, systems still matter. Let's say it's the zombie apocalypse. You'll stand a better chance of surviving if you use systems.

This includes not only systems of avoiding zombies. It also means systems of equations. A **system of equations** is a group of two or more equations with the same variables. There are many different kinds of word problems involving systems of equations. In this lesson, we'll focus on a few of the most common types of these problems. Remember, though, that the principles at work are the same in all of them.

Let's start with one that could save your life. It's the zombie apocalypse. You try to take shelter at Farmer Zed's place, but his barn is full of zombie animals. You know he had 20 animals, a mix of chickens and pigs. Your recon scout tells you he counts 50 legs, but can't see what they are. You hope there are more chickens, because zombie chickens are easier to kill. Zombie pigs? Not so much.

Okay, math can help. First, we need our variables. What don't we know? The number of chickens and pigs. Let's use *c* for chickens and *p* for pigs.

Next, we need our equations. We know *c* + *p* = 20. That's all the chickens and pigs combined. We know the chickens should have 2 legs and the pigs should have 4, supposing none were gnawed off by zombie ducks. So, 2*c* + 4*p* = 50.

To solve, let's rearrange this first equation to *c* = 20 - *p*, then substitute 20 - *p* for *c* in the second problem. This is the substitution method. So, that's 2(20 - *p*) + 4*p* = 50. We get 40 - 2*p* + 4*p* = 50. That becomes 2*p* = 10. So, *p* = 5.

Let's plug in 5 for *p* in *c* + *p* = 20. So, *c* = 15. Let's check our math by plugging *p* = 5 and *c* = 15 into the second equation. That's 2(15) + 4(5) = 50. 30 + 20 = 50. Okay, so 5 zombie pigs and 15 zombie chickens.

Let's try a problem involving money. In this post-apocalyptic hellscape you call home, you need two things most of all: bullets and cookies. Look, it may be the end times, but you still have a sweet tooth. Fortunately, you encounter some delusional retail workers who are convinced the monetary system should still function. I guess the zombies aren't much different than Black Friday shoppers to them.

Unfortunately, their cash registers no longer work, so you have no receipt. But, you know you spent $615.00 on a total of 135 $3.00 bullets and $8.00 cookies. How many of each did you get?

Let's make our variables *b* for bullets and *c* for cookies. That's good enough for me. What are our equations? You got 135 total items. So, *b* + *c* = 135. And, with a money problem, we can multiply the cost of each item times the number of the item to get its total cost. So, 3*b* is the cost of $3.00 bullets, and 8*c* is the cost of $8.00 cookies. You spent $615.00, so 3*b* + 8*c* = 615.

Let's use substitution again. We can make *b* + *c* = 135 into *b* = 135 - *c*. Then, substitute into the equation to get 3(135 - *c*) + 8*c* = 615. That simplifies to 405 - 3*c* + 8*c* = 615. 8*c* - 3*c* is 5*c*. And, 615 - 405 is 210. So, 5*c* = 210. Divide by 5, and get *c* = 42.

Plug 42 in for *c* in *b* + *c* = 135, and *b* = 93. Let's check for zombies. No zombies? Okay, let's check our work in the second equation. 3*b* + 8*c* = 615. That's 3(93) + 8(42) = 615. 3 * 93 is 279. 8 * 42 is 336. 279 + 336 is 615. So, we have 93 bullets and 42 cookies. Hmm, that might last a few days.

Let's try a percent problem. These can seem like tricky systems of equations questions, but they're really just using the same principles. Let's say you meet up with a chemist who has the key to stopping the zombification process. She needs a water-based solution with 14% vinegar. Yep, vinegar. Vinegar does all kinds of useful things.

Unfortunately, you only have vinegar in 8% and 24% solutions, and it only works at 14%. If you want 300 gallons of 14% solution, how much 8% and 24% will you need?

Let's use *x* and *y*. *x* can be the number of gallons we'll need of the 8% solution, and *y* is the same for the 24%. We know we want 300 gallons. So, *x* + *y* = 300. That's our first equation.

We also know we want the amount of vinegar in the 8% solution plus the amount in the 24% solution to equal 14% in the 300 gallons. We can rewrite what I just said as .08*x* + .24*y* = .14 * 300. That's the number of gallons of 8% times the 8%, which is .08, plus the number of *y* gallons times its concentration at 24%, equaling the final 300 gallons at 14 zombie-stopping percent.

Let's make that first equation *x* = 300 - y, and substitute it into the second equation to get .08(300 - *y*) + .24*y* = 42. Simplify that to 24 - .08*y* + .24*y* = 42. Let's get the *ys* together, and move the 24 over to get .16*y* = 18. Divide by .16 and get *y* = 112.5. Plug that into *x* + *y* = 300, and *x* = 187.5.

Again, check for zombies. Good? And cookies? Running a little low, but if we get this finished, more cookies await. Ok, let's plug 187.5 and 112.5 into our second equation. That's .08(187.5) + .24(112.5) = 42. .08 * 187.5 is 15. And, .24 * 112.5 is 27. 15 + 27? 42. Zombies? Vanquished!

In summary, a **system of equations** is simply a group of equations with the same variables. Problems involving systems of equations are useful whether you're organizing your yarn collection or battling the zombie threat. Just identify your variables, set up your equations, then solve for your variables. And, watch out for zombie ducks - they like to gnaw!

You should be able to solve word problems that involve setting up a system of equations after watching this video lesson.

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 79 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
9 in chapter 8 of the course:

Back To Course

ELM: CSU Math Study Guide15 chapters | 136 lessons

- What is a Linear Equation? 7:28
- How to Write a Linear Equation 8:58
- Problem solving using Linear Equations 6:40
- Solving Linear Equations with Literal Coefficients 5:40
- Solving Linear Equations: Practice Problems 5:49
- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- Solving a System of Equations with Two Unknowns 6:15
- Solving Problems Involving Systems of Equations 8:07
- Solving Linear Inequalities: Practice Problems 6:37
- Go to ELM Test - Algebra: Linear Equations & Inequalities

- PTE Academic Test: Practice & Study Guide
- TOEIC Listening & Reading: Test Prep & Practice
- Florida Supplemental Exam for Appraisers: Study Guide
- PFS Exam Study Guide - CPA Personal Financial Specialist
- Learning Calculus: Basics & Homework Help
- Castle History for Elementary School
- Planet Earth for Elementary School
- What Animals Eat for Elementary School
- Adaptations for Elementary School
- Aquatic Animal Adaptations for Elementary School
- Excelsior College BS in Business Degree Plan Using Study.com
- IELTS General Training Reading: Format & Task Types
- IELTS General Training Writing: Format & Task Types
- Gates-MacGinitie Reading Test Scores
- IELTS General Training Test: Structure & Scoring
- Supply and Demand Activities for Kids
- Speed Reading for Kids

- What Is the Uniform Commercial Code? - Definition & Example
- Cultural Globalization: Definition, Factors & Effects
- Effects of Fiscal & Monetary Policy on Personal Finance
- Effective Email Marketing Tips
- Information Technology: Roles & Responsibilities
- Following Directions Social Stories
- If-Then Activities for Kids
- Conglomerate Merger: Definition & Examples
- Quiz & Worksheet - Brabantio in Othello
- Quiz & Worksheet - FDR's First Inaugural Address
- Quiz & Worksheet - Using the Z Test Formula
- Quiz & Worksheet - What Is Gemeinschaft?
- Muscle Contraction Flashcards
- Water Polo Flashcards

- GRE Math Subject Test Study Guide & Test Prep
- Psychology 101: Intro to Psychology
- GED Math: Help and Review
- ILTS School Counselor: Test Practice and Study Guide
- Business 108: Business Ethics
- Qualitative Research Methods and Design: Help and Review
- Rational Functions and Difference Quotients: Tutoring Solution
- Quiz & Worksheet - Internet's Influence on Business Operations & Interactions
- Quiz & Worksheet - Normal Distribution & Shifts in the Mean
- Quiz & Worksheet - Capitalism vs. Pluralism vs. Democracy
- Quiz & Worksheet - Issues in Gender Equality in Education
- Quiz & Worksheet - Credit Terms on Invoices

- Differences Between Customers, Consumers & Consumerism
- 5 Feet 2 Inches in Meters: How-to & Steps
- Money Management Lesson Plan
- The Highwayman Lesson Plan
- 504 Plans in Arizona
- The SAT Essay: Scoring Scales & Reports
- 100th Day of School Project Ideas
- Collage Lesson Plan
- Weather Lesson Plan
- Common Core State Standards in Idaho
- 504 Plans in Massachusetts
- Parallel & Perpendicular Lines Lesson Plan

Browse by subject