Solving Linear Equations: A Step-by-Step Guide

by Tom Lembong 47 views
Iklan Headers

Hey there, math enthusiasts! Today, we're diving into the world of linear equations and tackling a classic problem: solving a system of two equations with two variables. We'll break down the process step by step, making it super easy to understand. So, grab your pencils and let's get started!

Linear equations are the backbone of many mathematical concepts, and understanding how to solve them is crucial. This particular problem presents us with a system of two equations: 2x + 3y = 8 and x - y = 1. Our goal is to find the values of x and y that satisfy both equations simultaneously. This means finding the point where the lines represented by these equations intersect on a graph. This is very important. To solve this, we will use the substitution method or the elimination method. Don't worry, both are pretty straightforward, so lets do it.

Understanding the Problem: The Core of Solving Linear Equations

Before we jump into the methods, let's make sure we're all on the same page. The system of equations we're working with is:

  • 2x + 3y = 8
  • x - y = 1

Each of these equations represents a straight line when graphed. The solution to the system is the point (x, y) where these two lines cross each other. This point's coordinates, when substituted into both equations, will make both equations true. It's like finding a treasure that fits perfectly in two different maps, each giving you a different clue. This is why it is very important. The solutions must satisfy the two equations.

Think of x and y as unknowns. Our job is to find the exact values these variables must have for both equations to be correct at the same time. Remember that each equation is like a constraint. So the variables should satisfy the condition of the two equations. In other words, to solve a system of linear equations, you are looking for values of the variables that satisfy all equations in the system. The solution represents the point(s) where the graphs of the equations intersect.

Now, let's walk through the steps, ensuring you grasp the 'why' behind each move.

Method 1: The Substitution Method – Unveiling the Values

The substitution method is like solving a puzzle piece by piece. Here’s how it works:

  1. Isolate a Variable: Look at the equations and choose one where it's easiest to isolate a variable. In our case, the second equation (x - y = 1) is perfect because x is simple to isolate. We can rewrite this as: x = y + 1. We just need to move y to the other side of the equation. This gives us an expression for x in terms of y.
  2. Substitute: Now, take this expression for x (which is y + 1) and substitute it into the other equation (2x + 3y = 8). Replace every instance of x with (y + 1). So, the first equation becomes 2(y + 1) + 3y = 8.
  3. Solve for the Remaining Variable: Simplify and solve the new equation for y. Expanding the equation, we get 2y + 2 + 3y = 8. Combine like terms (2y and 3y) to get 5y + 2 = 8. Subtract 2 from both sides: 5y = 6. Finally, divide by 5: y = 6/5. So, we've found the value of y.
  4. Find the Other Variable: Now that you know y = 6/5, plug this value back into either of the original equations or the modified equation (x = y + 1) to find x. Using x = y + 1, we get x = 6/5 + 1. Converting 1 to a fraction with a denominator of 5, we get x = 6/5 + 5/5, which simplifies to x = 11/5. That's it! Now we know x and y values. The solution to the system of equations is (11/5, 6/5).

Method 2: The Elimination Method – Canceling Out Variables

The elimination method, also known as the addition method, is all about canceling out one of the variables. Here’s how it goes:

  1. Prepare the Equations: The goal is to get the coefficients of either x or y to be opposites (e.g., +3 and -3). In our example, we can easily make the y coefficients opposites. Multiply the second equation (x - y = 1) by 3. This changes the equation to 3x - 3y = 3.
  2. Eliminate a Variable: Now, add the modified second equation (3x - 3y = 3) to the first equation (2x + 3y = 8). This gives us: (2x + 3x) + (3y - 3y) = 8 + 3. The y terms cancel out! This simplifies to 5x = 11.
  3. Solve for the Remaining Variable: Divide both sides by 5: x = 11/5. We have found the x value.
  4. Find the Other Variable: Substitute x = 11/5 into either of the original equations. Let’s use x - y = 1. We get 11/5 - y = 1. Rewrite 1 as 5/5, so 11/5 - y = 5/5. Subtract 11/5 from both sides: -y = 5/5 - 11/5, which gives -y = -6/5. Multiply by -1: y = 6/5. And we're done! The solution is (11/5, 6/5). The solution satisfies the original equations.

Comparing Methods

The substitution method is great when one of the equations is already solved for a variable or is easy to rearrange. The elimination method shines when the equations are neatly aligned, and you can easily make the coefficients of one of the variables opposites.

Final Answer and Explanation

Alright guys, the solutions we obtained for x and y using both the substitution and elimination methods are x = 11/5 and y = 6/5. Let's analyze the multiple-choice options:

  • A) x = 2 and y = 1: Let’s plug these values into the original equations. For the first equation, 2(2) + 3(1) = 4 + 3 = 7, which does not equal 8. So this is not correct.
  • B) x = 1 and y = –1: Substituting these values into the first equation, we get 2(1) + 3(-1) = 2 - 3 = -1, which is not equal to 8. This isn't the correct answer either.
  • C) x = 8 and y = –1: Substituting, 2(8) + 3(-1) = 16 - 3 = 13, which is not equal to 8. This is not the correct solution.
  • D) x = 3 and y = 0: Substituting these values into the first equation, we have 2(3) + 3(0) = 6, which is not equal to 8. Not the correct solution.

None of the multiple-choice options match the correct solution that we calculated, which is x = 11/5 and y = 6/5. The provided options are incorrect. Always double-check your work, and remember, practice makes perfect! Keep solving these equations, and you'll become a pro in no time.

Conclusion: Mastering the Art of Solving Linear Equations

So there you have it – a clear, step-by-step guide to solving systems of linear equations. We’ve covered both the substitution and elimination methods, armed with the knowledge to tackle any problem that comes your way. Remember, the key is to choose the method that best fits the problem and to always double-check your work. Practice these techniques, and you'll gain confidence. Keep practicing, and you will become proficient.

Keep exploring, keep learning, and don't be afraid to make mistakes – that's how we grow! Until next time, happy solving!