Algebra 1: Systems of Equations Word Problems

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Questions and Answers

How do you set up and solve a system of equations word problem?

Define your variables, follow the wording to create equations, and use elimination or substitution.

What are the two numbers if their sum is 12 and their difference is 4?

4 and 8

What are the two numbers if their difference is 3 and their sum is 13?

5 and 8

How many fancy shirts and plain shirts did Kristin buy if she spent $131, with fancy shirts at $28 each and plain shirts at $15 each?

2 fancy shirts and 5 plain shirts Signup and view all the answers

How many chickens and pigs are there if there are 13 animals and a total of 40 legs?

6 chickens and 7 pigs Signup and view all the answers

How many ducks and pigs are there if a farmhouse shelters 10 animals and has 36 legs?

2 ducks and 8 pigs Signup and view all the answers

How many cars and buses were taken by 195 students in 7 vehicles?

3 cars and 4 buses Signup and view all the answers

How many vans and buses did the Math Club use if 231 students are in total and they used 15 vehicles?

8 vans and 7 buses Signup and view all the answers

How many Model A and Model B printing presses does Elisa's company have if they own a total of 14 presses and can print 905 books per day?

9 of Model A and 5 of Model B Signup and view all the answers

How many forks and spoons did Molly's Custom Kitchen Supplies make if they spent $37.90 on materials and sold them for $147.20?

7 forks and 20 spoons Signup and view all the answers

How many medals did Erin buy if she spent $1,400 on 50 items at $25 each for medals and $35 each for trophies?

35 medals Signup and view all the answers

Which of the following systems gives the price of a bagel (b) and coffee (c) if Oleg buys 3 bagels and 2 cups for $13.50?

3b + 2c = 13.50, b = 2 + c Signup and view all the answers

What is the value of $xy$ if the sum of $x$ and $y$ is 12 and $y$ is half of $x$?

32 Signup and view all the answers

Which system of equations represents Yisel's order for buckets of paint containing a total of 100 gallons?

s + m = 40, s + 3m = 100 Signup and view all the answers

How many express passes did the group purchase if they paid $432 for twenty 7-day passes?

12 Signup and view all the answers

What are the values of $r$ and $s$ in the expression $rx + s$ for local pass sales?

r = -5, s = 60 Signup and view all the answers

What is the area of the full rectangle if the area of a circle is 4 times that of the rectangle?

10 Signup and view all the answers

What is the value of $b$ if $ac = 135$, $a$ is 3 less than $b$, and $c$ is 6 more than $a$?

12 Signup and view all the answers

Study Notes

Setting Up Systems of Equations

Define variables at the beginning (e.g., let x = first number, y = second number).
Create equations based on problem statements using addition, subtraction, etc.
Generally results in at least two equations to solve.

Common Word Problems and Solutions

Finding Two Numbers:
- For problems such as "Their sum is 12" and "Their difference is 4", derive:
  - ( x + y = 12 )
  - ( x - y = 4 )
- Solutions can be found using elimination or substitution techniques.
Shirts Purchase Problem:
- Define variables for types of shirts (e.g., ( f ) for fancy shirts, ( p ) for plain shirts).
- Example equations:
  - Total shirts: ( f + p = 7 )
  - Total cost: ( 28f + 15p = 131 )
- Use substitution to find the number of each shirt type.
Animal Count Problems:
- For total animal counts and leg counts (e.g., "13 animals" and "40 legs"), set up equations based on counts of chickens and pigs.
Vehicle Transportation Problems:
- Incorporate total numbers of vehicles and the capacity of each type to devise equations. Use factors like passenger capacity to find the number of different vehicles taken on trips.
Printing Press Problem:
- Define variables for models of printing presses and derive equations based on total numbers and output capacity.
Kitchen Supplies Financial Problem:
- Determine quantities of products sold based on costs and sales revenue to formulate equations.
Medals and Trophies Purchasing Problem:
- Use cost equations based on quantities purchased to establish systems that solve for specific numbers of items bought.
Bakery Pricing Problem:
- Formulate equations around the costs of items sold, ensuring to relate the prices of bagels and coffee.

Advanced Problems

Express Train Passes:
- Involves equations based on collective spending for multiple passes to calculate quantities purchased.
Circle and Rectangle Area Problem:
- Establish relationships between areas of geometric shapes, working with conditions involving their overlaps.
Integer Relationships:
- For multiple variable systems derived from integer relationships (like ( a ), ( b ), ( c )), derive equations based on given conditions and solve for specific integers.

Overall, a solid understanding of forming equations from word problems will facilitate finding solutions in algebraic contexts. Aim for clarity in defining variables and consistency in your mathematical setups.

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Description

This quiz focuses on solving word problems involving systems of equations. You'll learn to define variables, translate problem statements into equations, and apply methods like elimination and substitution. Enhance your understanding of setting up and solving these mathematical challenges.