Solving Linear Equations A Step-by-Step Guide With Henrique

Hey guys! Let's dive into a cool math problem together. We're going to break down how Henrique tackled a system of linear equations using the linear combination method. It's like detective work with numbers, and we'll make sure every step makes perfect sense. So, grab your thinking caps, and let's get started!

The Problem Unveiled

Our mission begins with the system of equations Henrique faced. He had to solve:

4x - 7y = 28
6x - 5y = 31

To crack this, Henrique chose the linear combination method, a slick way to eliminate one variable by making the coefficients of either x or y opposites. This sets us up for some sweet cancellations and a simpler equation to solve.

Step 1: Setting the Stage for Elimination

Henrique's first move was to multiply each equation by a strategic number. He aimed to make the x coefficients opposites. Here’s what he did:

1. Multiply the first equation by 3:

   3(4x - 7y) = 3(28)
   

   This gives us:

   12x - 21y = 84
   

2. Multiply the second equation by -2:

   -2(6x - 5y) = -2(31)
   

   Which results in:

   -12x + 10y = -62
   

So, Henrique transformed the original system into:

12x - 21y = 84
-12x + 10y = -62

Notice how the x terms are now 12x and -12x? Perfect opposites, just as planned! This strategic multiplication is a key part of the linear combination method, ensuring we can eliminate a variable in the next step. By focusing on aligning the coefficients, Henrique set the stage for a smooth solution, making the equations easier to handle and bringing us closer to finding the values of x and y.

Step 2: The Elimination Act

Now for the magic trick! Henrique added the two modified equations together. This is where the linear combination method truly shines.

(12x - 21y) + (-12x + 10y) = 84 + (-62)

When we combine like terms, the 12x and -12x cancel each other out—poof! This leaves us with:

-11y = 22

Just like that, we've eliminated x and have a straightforward equation with just y. This step is crucial because it simplifies the problem, allowing us to solve for one variable at a time. The beauty of this method is how it transforms a complex system into a manageable single-variable equation. Henrique's skillful setup in the previous step made this elimination possible, turning a seemingly tough problem into a clear path toward the solution. By focusing on eliminating one variable, we’re one giant leap closer to finding the values of both x and y.

Step 3: Unmasking 'y'

With the equation -11y = 22, Henrique is on the home stretch to uncover the value of y. To isolate y, he simply divides both sides of the equation by -11:

y = 22 / -11

This calculation reveals:

y = -2

Bravo! Henrique has successfully found the value of y. This step is a prime example of how algebraic manipulation can simplify equations and bring us closer to the solution. By performing this straightforward division, Henrique has cleared a significant hurdle in solving the system. Knowing the value of y is a pivotal moment, as it sets the stage for finding x. It’s like finding a key piece of a puzzle that unlocks the rest of the solution. With y in hand, we’re ready to move on to the final act: discovering the value of x.

Step 4: The Quest for 'x'

Now that Henrique knows y = -2, he can find x. He picks one of the original equations—let's go with 4x - 7y = 28—and plugs in the value of y:

4x - 7(-2) = 28

This simplifies to:

4x + 14 = 28

Next, he subtracts 14 from both sides:

4x = 14

Finally, Henrique divides by 4 to isolate x:

x = 14 / 4

Which simplifies to:

x = 7 / 2

Or, as a decimal:

x = 3.5

Fantastic! Henrique has pinpointed the value of x. This substitution step is a classic move in solving systems of equations, showcasing how the value of one variable can unlock the other. By carefully plugging y into the equation, Henrique transformed it into a simple equation with just x, making it easy to solve. Finding x completes our mission, providing the final piece of the puzzle. With both x and y in hand, we've successfully navigated the system of equations and can confidently present the solution.

Step 5: Victory Lap – The Solution Unveiled

After all the calculations, Henrique arrives at the solution. The values that satisfy both equations are:

x = 3.5
y = -2

We often write this as an ordered pair: (3.5, -2). This ordered pair is the ultimate answer, the point where the two lines represented by the equations intersect on a graph. Checking this solution by plugging these values back into the original equations is a smart move to ensure everything checks out. This final step solidifies the solution and gives us confidence in our answer. The solution represents a harmonious balance, where both equations hold true, confirming our successful journey through the problem. Congrats to Henrique for cracking the code!

Common Pitfalls and How to Dodge Them

Solving systems of equations can be tricky, and it’s easy to stumble along the way. Here are some common mistakes and how to avoid them:

• Sign Slip-Ups: A very common mistake is messing up the signs when multiplying or adding equations. Always double-check your signs! When distributing a negative number, ensure every term inside the parentheses gets the correct sign. Pay extra attention during the addition step to avoid overlooking sign changes.
• Multiplication Mishaps: Make sure you multiply every term in the equation when you're scaling them. It's super easy to forget a term, especially if you're working quickly. Double-check that each term on both sides of the equation has been correctly multiplied to maintain the equation's balance.
• Substitution Snafus: When substituting, be extra careful to replace the correct variable and use parentheses to handle signs correctly. Always double-check which variable you are substituting and ensure the substituted value is placed correctly. Parentheses are your friends, especially when dealing with negative numbers, as they help maintain the correct sign.
• Arithmetic Antics: Simple arithmetic errors can throw off the whole solution. Take your time and double-check every calculation. Even seasoned mathematicians can make mistakes, so it’s always wise to review your arithmetic. Using a calculator for complex calculations can minimize these errors.
• Forgetting the Final Flourish: Don't forget to solve for both x and y. It's easy to get so focused on finding one variable that you forget the other. Always remember to back-substitute your found value to solve for the remaining variable. This ensures you have the complete solution to the system.

By being mindful of these common pitfalls and taking the time to double-check each step, you can boost your accuracy and confidence in solving systems of equations. Happy solving!

Linear Combination Mastery: Tips and Tricks

To truly master the linear combination method, here are some extra tips and tricks that can help you tackle any system of equations with confidence:

• Strategic Selection: Before diving in, take a moment to assess the equations. Look for the easiest way to eliminate a variable. Sometimes, one variable might have coefficients that are easier to manipulate. For example, if one equation has a 2y and the other has a -y, you can simply multiply the second equation by 2 to eliminate y. Choosing the right variable to eliminate first can save you steps and reduce the chance of errors.
• LCM Power: When coefficients aren't straightforward, use the Least Common Multiple (LCM) to find the smallest numbers to multiply by. This keeps the numbers manageable and reduces the risk of mistakes. For instance, if you have equations with 3x and 5x, the LCM of 3 and 5 is 15, so you would aim to make the x coefficients 15x and -15x. Using LCM ensures you’re working with the smallest multiples necessary, simplifying the arithmetic.
• Flexibility is Key: Sometimes, you might need to multiply both equations to eliminate a variable. Don't hesitate to do this if it simplifies the problem. This is especially useful when the coefficients don't share a simple multiple. For example, if you have 2x + 3y = 7 and 3x - 4y = 10, you might multiply the first equation by 3 and the second by -2 to eliminate x.
• Fraction Frustration Fix: If you encounter fractions, clear them out early by multiplying the entire equation by the denominator. This makes the equation much easier to work with. For example, if you have an equation like (1/2)x + (2/3)y = 5, multiply the whole equation by 6 (the LCM of 2 and 3) to get rid of the fractions.
• Verification Victory: Always, always, always check your solution by plugging the values of x and y back into the original equations. This ensures you haven't made any mistakes along the way. If the solution doesn't satisfy both equations, you know there’s an error somewhere, and you can go back and review your steps.

By incorporating these tips and tricks into your problem-solving toolkit, you'll become a linear combination master in no time! Keep practicing, and you'll find these methods become second nature. Happy equation solving!

Real-World Revelations: Linear Equations in Action

Linear equations aren't just abstract math problems; they pop up all over the real world! Understanding how to solve them can be super useful in various situations. Let’s explore some cool real-world applications where linear equations shine:

• Budgeting Brilliance: Ever tried to figure out how to spend your money wisely? Linear equations can help you create a budget! For example, if you have a fixed income and different expenses, you can set up equations to see how much you can spend on each category while staying within your budget. Let’s say you earn $2000 a month and want to allocate funds for rent, groceries, and entertainment. You can create an equation like Rent + Groceries + Entertainment = 2000 and use linear equations to find the optimal amounts for each category.
• Mixing Solutions Magic: Chemists and pharmacists often use linear equations to determine the amounts of different solutions needed to create a specific concentration. This is crucial for creating medications and conducting experiments. For instance, if you need to mix a 20% saline solution with a 50% saline solution to get 100 mL of a 30% solution, linear equations can help you figure out exactly how much of each to use.
• Travel Time Tactics: Planning a road trip? Linear equations can help you estimate travel times and distances. If you know the distance you need to travel and your average speed, you can use the formula Distance = Speed × Time to calculate how long the trip will take. You can also use systems of equations to account for different legs of the journey with varying speeds and distances.
• Investment Insights: Linear equations are valuable tools in finance. They can help you calculate simple interest, analyze investment returns, and compare different investment options. For example, you can use linear equations to determine how long it will take for an investment to reach a certain value at a given interest rate. This helps in making informed decisions about savings and investments.
• Sales and Discounts Savvy: Shopping for a bargain? Linear equations can help you calculate discounts and sale prices. If an item is 20% off, you can use a linear equation to find the sale price. Understanding these calculations ensures you’re getting the best deals and helps you manage your spending effectively.

These real-world examples highlight how linear equations are more than just numbers and symbols on a page. They’re powerful tools that help us solve practical problems and make informed decisions in everyday life. So, next time you encounter a real-world challenge, remember that linear equations might just be the key to unlocking the solution!

Wrapping up, Henrique’s journey through the system of equations perfectly illustrates the power and elegance of the linear combination method. By strategically eliminating variables, we transformed a complex problem into a manageable solution. We also explored common pitfalls, shared tips and tricks, and discovered the real-world applications of linear equations. So, keep practicing, keep exploring, and embrace the problem-solving magic of mathematics! You've got this!