Solving 3x - 4y = 55 A Comprehensive Guide

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Hey guys! Ever stumbled upon a math problem that looks like it’s speaking another language? You're definitely not alone! Equations like 3x - 4y = 55 can seem intimidating at first glance, but trust me, once we break it down, it's totally manageable. In this article, we're diving deep into this equation, exploring what it means, and how we can find the solutions. We'll cover everything from the basics of linear equations to practical methods for solving them. So, buckle up, grab your thinking caps, and let’s get started!

Understanding the Equation: 3x - 4y = 55

Let's begin by decoding what this equation actually represents. The equation 3x - 4y = 55 is a linear equation with two variables, x and y. Now, what does that mean? Simply put, it describes a straight line when graphed on a coordinate plane. Each point on that line represents a pair of x and y values that make the equation true. Think of it like this: you have two unknowns (x and y), and they are related in a specific way. This equation tells us exactly how they are related. The coefficients (the numbers in front of the variables) and the constant (the number on the right side of the equation) define the slope and position of the line.

What Makes It a Linear Equation?

A linear equation has a few key characteristics:

  1. Variables to the First Power: Notice that both x and y are raised to the power of 1. There are no exponents like x^2 or square roots of y. This is crucial because it ensures the relationship is linear – meaning it forms a straight line.
  2. No Multiplication of Variables: You won't see terms like xy in a linear equation. The variables are added or subtracted, but never multiplied together.
  3. Constant Coefficients: The numbers multiplying x and y (3 and -4 in this case) are constants. They don't change, which helps keep the relationship consistent.

The Significance of Two Variables

The fact that we have two variables means that there isn't just one single solution. Instead, there are infinitely many pairs of x and y that satisfy the equation. Each pair represents a point on the line. This is different from an equation with only one variable (like 3x = 9), which has a unique solution (x = 3). With two variables, we're dealing with a range of solutions that form a line.

Visualizing the Equation

Imagine a coordinate plane with an x-axis and a y-axis. The equation 3x - 4y = 55 represents a line drawn on this plane. Every point on this line has coordinates (x, y) that, when plugged into the equation, make it true. Some points will be easy to spot, while others might be fractions or decimals. The beauty of a linear equation is that it provides a clear, visualizable relationship between two variables.

Practical Applications

Linear equations aren't just abstract math concepts; they show up in tons of real-world situations. For example, they can model relationships between:

  • The cost of items and the number you buy
  • The distance traveled and the time it takes
  • The temperature in Celsius and Fahrenheit

Understanding how to work with linear equations is a fundamental skill that opens doors to more advanced math and problem-solving in various fields.

Finding Solutions: Methods and Approaches

Okay, now that we understand what the equation means, let's talk about how to find solutions. Since there are infinitely many solutions, we can't list them all. Instead, we can use different methods to find specific solutions or to express the general relationship between x and y. Here are a few common approaches:

1. Substitution Method

The substitution method involves solving the equation for one variable in terms of the other and then plugging that expression back into the equation. This gives us a way to find specific solutions by choosing a value for one variable and calculating the other.

Step-by-Step Example:

  1. Solve for One Variable: Let's solve the equation 3x - 4y = 55 for x. We get:
    3x = 4y + 55
x = (4y + 55) / 3
  2. Choose a Value for y: Now, we can pick any value for y and plug it into the equation to find the corresponding x. For example, let's choose y = 1. This is a critical step, guys, because it allows us to pinpoint a specific solution pair.
  3. Calculate x: Substitute y = 1 into the equation:
    x = (4(1) + 55) / 3
x = (4 + 55) / 3
x = 59 / 3
    So, when y = 1, x = 59/3 (approximately 19.67). This gives us one solution: (59/3, 1). Understanding how to execute this calculation effectively is key.
  4. Find More Solutions: We can repeat this process with different values of y to find as many solutions as we need. For instance, if we choose y = 4:
    x = (4(4) + 55) / 3
x = (16 + 55) / 3
x = 71 / 3
    Another solution is (71/3, 4). It's fascinating how each choice for 'y' opens the door to a new solution.

2. Isolating a Variable

Another effective technique is isolating one of the variables to express it in terms of the other. This is similar to the first step in the substitution method but can be useful on its own for understanding the relationship between the variables.

Step-by-Step Example:

  1. Isolate x: As we did before, we can solve the equation for x:
    3x = 4y + 55
x = (4y + 55) / 3
    This equation tells us that x is equal to (4y + 55) / 3. We've successfully expressed x in terms of y.
  2. Analyze the Relationship: This form of the equation makes it clear how x changes as y changes. For every increase in y, x will also change in a predictable way. This insight into the interdependence of x and y is valuable.
  3. Create Solution Pairs: By choosing different values for y, we can easily calculate the corresponding values for x. This method efficiently generates solution pairs.

3. Graphing the Equation

Visually representing the equation can provide a deep understanding of its solutions. The graph of 3x - 4y = 55 is a straight line, and every point on that line corresponds to a solution.

Step-by-Step Example:

  1. Find Two Points: To graph a line, we need at least two points. We can use the substitution method or isolating a variable to find these points. Let's use two solutions we found earlier: (59/3, 1) and (71/3, 4). These points will serve as anchors for our line.
  2. Plot the Points: Plot these points on a coordinate plane. Make sure to accurately place them according to their x and y coordinates.
  3. Draw the Line: Draw a straight line through the two points. This line represents all the solutions to the equation 3x - 4y = 55. The precision with which we draw the line will determine how accurately we can read off solutions.
  4. Read Solutions from the Graph: Any point on this line represents a solution to the equation. By looking at the graph, we can estimate other solutions. For example, if we find a point on the line where y = 7, we can visually approximate the corresponding x value. The beauty of this method lies in its visual nature, allowing us to 'see' the infinite solutions.

Practical Tips for Solving

  • Choose Smart Values: When using the substitution method, try choosing values for y that will make the calculations easier. For example, if the equation involves fractions, picking values that cancel out the denominator can simplify things.
  • Check Your Answers: Always plug your solutions back into the original equation to make sure they work. This helps prevent errors and builds confidence in your results.
  • Use Online Tools: There are many online calculators and graphing tools that can help you solve and visualize linear equations. These tools can be especially useful for checking your work or exploring different solutions.

Real-World Applications of Linear Equations

Linear equations aren't just abstract math problems; they're powerful tools for modeling and solving real-world situations. Understanding how to apply these equations can help us make informed decisions and solve practical problems.

Example 1: Budgeting and Spending

Let's say you have a budget of $200 per month for entertainment. You want to see movies and go bowling. Movie tickets cost $12 each, and bowling costs $10 per game. We can set up a linear equation to represent this situation.

  • Let x be the number of movie tickets you buy.
  • Let y be the number of bowling games you play.

The equation representing your budget constraint is:

12x + 10y = 200

This equation tells us that the total cost of movies and bowling must equal your budget of $200. We can use this equation to figure out different combinations of movies and bowling games you can enjoy within your budget.

Solving the Problem:

  1. Rearrange the equation: Let's solve for y to express the number of bowling games in terms of movie tickets:

    10y = 200 - 12x
y = (200 - 12x) / 10
y = 20 - 1.2x

  2. Find possible solutions: Now, we can plug in different values for x (number of movie tickets) and calculate the corresponding values for y (number of bowling games). Since we can't buy a fraction of a movie ticket or play a fraction of a bowling game, we'll only consider whole numbers.

    • If x = 0 (no movies), then y = 20 - 1.2(0) = 20. You can play 20 bowling games.
    • If x = 5 (5 movies), then y = 20 - 1.2(5) = 20 - 6 = 14. You can play 14 bowling games.
    • If x = 10 (10 movies), then y = 20 - 1.2(10) = 20 - 12 = 8. You can play 8 bowling games.
    • If x = 15 (15 movies), then y = 20 - 1.2(15) = 20 - 18 = 2. You can play 2 bowling games.
    • If x = 16 (16 movies), then y = 20 - 1.2(16) = 20 - 19.2 = 0.8. This isn't a whole number, so it's not a practical solution.

  3. Interpret the solutions: We've found several possible combinations:

    • 0 movies and 20 bowling games
    • 5 movies and 14 bowling games
    • 10 movies and 8 bowling games
    • 15 movies and 2 bowling games

    You can choose the combination that best fits your preferences while staying within your budget. This approach effectively illustrates the power of math in managing our financial decisions.

Example 2: Distance, Rate, and Time

Suppose you're planning a road trip. You want to drive 300 miles, and you want to know how the driving time changes with different average speeds. We can use a linear equation to model this situation.

The formula relating distance, rate, and time is:

Distance = Rate × Time

In this case, the distance is fixed at 300 miles. Let x be the rate (average speed in miles per hour), and let y be the time (in hours). The equation becomes:

300 = xy

This isn't a linear equation in the traditional ax + by = c form, but we can still analyze the relationship between x and y.

Solving the Problem:

  1. Rearrange the equation: Solve for y to express time in terms of speed:

    y = 300 / x

  2. Find possible solutions: Now, we can plug in different values for x (average speed) and calculate the corresponding values for y (driving time).

    • If x = 50 mph, then y = 300 / 50 = 6 hours.
    • If x = 60 mph, then y = 300 / 60 = 5 hours.
    • If x = 75 mph, then y = 300 / 75 = 4 hours.

  3. Interpret the solutions: We've found that:

    • Driving at 50 mph will take 6 hours.
    • Driving at 60 mph will take 5 hours.
    • Driving at 75 mph will take 4 hours.

    This equation helps you understand how your driving time decreases as your speed increases. Understanding this relationship can be invaluable in planning our travels effectively.

Example 3: Mixture Problems

Imagine you're a chemist mixing two solutions with different concentrations of a certain chemical. You need to create a specific concentration by combining the solutions.

Let's say you have two solutions: Solution A is 20% chemical, and Solution B is 50% chemical. You want to create 100 liters of a solution that is 30% chemical.

  • Let x be the amount (in liters) of Solution A.
  • Let y be the amount (in liters) of Solution B.

We can set up two equations:

  1. Total volume: The total volume of the mixture is 100 liters:
    x + y = 100
  2. Chemical concentration: The amount of chemical in the mixture is 30% of 100 liters, which is 30 liters:
    0.20x + 0.50y = 30

Now we have a system of two linear equations with two variables.

Solving the Problem:

  1. Solve the first equation for one variable: Let's solve for x:
    x = 100 - y
  2. Substitute into the second equation: Substitute this expression for x into the second equation:
    0.  20(100 - y) + 0.50y = 30
20 - 0.20y + 0.50y = 30
0.  30y = 10
y = 10 / 0.30
y ≈ 33.33
  3. Solve for the other variable: Substitute the value of y back into the equation for x:
    x = 100 - 33.33
x ≈ 66.67
  4. Interpret the solutions: You need approximately 66.67 liters of Solution A and 33.33 liters of Solution B to create 100 liters of a 30% chemical solution. This application demonstrates the versatility of linear equations in addressing real-world challenges in science and industry.

Conclusion: The Power of Linear Equations

So, guys, we've journeyed through the world of linear equations, taking the equation 3x - 4y = 55 as our guide. We've seen what it means, how to find its solutions, and how these equations pop up in everyday situations. Linear equations are like the basic building blocks of math – they might seem simple, but they're incredibly powerful. From budgeting to planning trips to mixing chemicals, they help us make sense of the world around us. Keep practicing, keep exploring, and you'll find that these equations become your trusty tools for problem-solving!