System Of Equations: Translate Word Problems!

Hey guys! Let's dive into the exciting world of translating word problems into systems of equations. It's like turning a riddle into a clear, solvable puzzle! Today, we're tackling a classic example that will help you master this skill. So, grab your thinking caps, and let’s get started!

Understanding the Basics

Before we jump into the problem, let's quickly recap what a system of equations is. A system of equations is simply a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Think of it as finding the common ground where all the equations agree.

Now, translating word problems into equations involves identifying the key information and representing it using mathematical symbols. This usually means assigning variables to unknown quantities and then writing equations that express the relationships described in the problem. It's like being a detective, piecing together clues to solve the mystery!

Breaking Down the Problem

Let’s consider our specific problem: We have two numbers, and we're going to call them ${ x }$ and ${ y }$. The problem gives us two pieces of information:

1. Three times the first number decreased by the second number is -5.
2. The first number increased by twice the second number is 17.

Our mission, should we choose to accept it, is to convert these sentences into mathematical equations. Ready? Let's do this!

Translating the First Sentence

The first sentence is: "Three times the first number decreased by the second number is -5." Let's break it down step by step:

• "Three times the first number" translates to ${ 3x }$.
• "Decreased by the second number" means we subtract ${ y }$, so we have ${ 3x - y }$.
• "Is -5" tells us that the expression equals -5. Therefore, the first equation is: ${3x - y = -5}$

See? That wasn't so bad! We've successfully transformed the first sentence into a neat little equation.

Translating the Second Sentence

Now, let's tackle the second sentence: "The first number increased by twice the second number is 17."

• "The first number" is simply ${ x }$.
• "Increased by twice the second number" means we add ${ 2y }$, so we have ${ x + 2y }$.
• "Is 17" tells us that the expression equals 17. Therefore, the second equation is: ${x + 2y = 17}$

Boom! We've done it again. The second sentence is now a beautiful equation, ready to be solved.

The System of Equations

Now that we've translated both sentences, we can write the complete system of equations:

${ \begin{cases} 3x - y = -5 \\ x + 2y = 17 \end{cases} }$

This system of equations represents the relationships described in the original word problem. To solve this system, you can use methods like substitution or elimination to find the values of ${ x }$ and ${ y }$ that satisfy both equations.

Solving the System (Optional)

Just for fun, let's go ahead and solve this system using the substitution method. First, we'll solve the second equation for ${ x }$: ${x = 17 - 2y}$ Now, substitute this expression for ${ x }$ into the first equation: ${3(17 - 2y) - y = -5}$ Expand and simplify: ${51 - 6y - y = -5}$ ${51 - 7y = -5}$ ${-7y = -56}$ ${y = 8}$ Now that we have ${ y = 8 }$, we can substitute it back into the equation ${ x = 17 - 2y }$: ${x = 17 - 2(8)}$ ${x = 17 - 16}$ ${x = 1}$ So, the solution to the system of equations is ${ x = 1 }$ and ${ y = 8 }$.

Why This Matters

You might be wondering, "Why do I need to learn this?" Well, translating word problems into systems of equations is a fundamental skill in mathematics and has applications in various fields. Here are a few reasons why it's important:

1. Problem-Solving: It helps you break down complex problems into smaller, manageable parts.
2. Real-World Applications: Many real-world scenarios can be modeled using systems of equations, such as determining the optimal mix of products to maximize profit or calculating the trajectory of a projectile.
3. Critical Thinking: It enhances your critical thinking and analytical skills, as you need to identify the relationships between different quantities.
4. Foundation for Advanced Math: It lays the groundwork for more advanced mathematical concepts, such as linear algebra and calculus.

Tips for Success

Here are some tips to help you become a pro at translating word problems into systems of equations:

• Read Carefully: Read the problem multiple times to make sure you understand all the information.
• Identify the Unknowns: Determine what quantities you need to find and assign variables to them.
• Look for Keywords: Pay attention to keywords like "sum," "difference," "times," and "is," as they indicate mathematical operations.
• Write Equations: Translate the sentences into equations using the variables you defined.
• Check Your Work: After solving the system, plug the values back into the original equations to make sure they are satisfied.
• Practice, Practice, Practice: The more you practice, the better you'll become at translating word problems into equations.

Common Mistakes to Avoid

Even the best of us make mistakes sometimes! Here are some common pitfalls to watch out for:

• Misinterpreting Keywords: Be careful with keywords like "less than" or "more than," as they can be tricky. For example, "5 less than x" is ${ x - 5 }$, not ${ 5 - x }$.
• Incorrectly Assigning Variables: Make sure you assign variables to the correct quantities. For example, if the problem asks for the number of apples and oranges, don't assign ${ x }$ to the total number of fruits.
• Not Checking Your Work: Always check your solution by plugging the values back into the original equations. This will help you catch any mistakes you may have made.
• Rushing Through the Problem: Take your time and read the problem carefully. Rushing can lead to mistakes.

Let's Practice!

Now that we've covered the basics, let's try a few more examples to solidify your understanding. Remember, practice makes perfect!

Example 1

The sum of two numbers is 25, and their difference is 5. Find the numbers.

Let ${ x }$ be the first number and ${ y }$ be the second number. The system of equations is:

${ \begin{cases} x + y = 25 \\ x - y = 5 \end{cases} }$

Example 2

A collection of dimes and quarters is worth $5.50. There are 22 coins in total. How many dimes and quarters are there?

Let ${ d }$ be the number of dimes and ${ q }$ be the number of quarters. The system of equations is:

${ \begin{cases} 0. 10d + 0.25q = 5.50 \\ d + q = 22 \end{cases} }$

Conclusion

Alright, guys, that's a wrap! You've learned how to translate word problems into systems of equations, a valuable skill that will serve you well in mathematics and beyond. Remember to read carefully, identify the unknowns, and practice regularly. With a little effort, you'll be solving systems of equations like a pro in no time!

Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this! Now go out there and conquer those word problems!