Solving Systems Of Equations: A Step-by-Step Guide

by Andrew McMorgan 51 views

Hey guys! Today, we're diving into the fascinating world of systems of equations. Specifically, we'll be tackling the system: -3x + 4y = 12 and (1/4)x - (1/3)y = 1. We'll break down the steps to find the solution, making it super easy to understand. So, grab your pencils and let's get started!

Understanding Systems of Equations

Before we jump into solving, let's quickly chat about what a system of equations actually is. Basically, it's a set of two or more equations that share the same variables. Our goal is to find the values of those variables that make all the equations true at the same time. Think of it like finding the perfect meeting point for multiple lines on a graph.

There are a few ways we can go about solving these systems. The two main methods are:

  • Substitution: We solve one equation for one variable and then substitute that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining one.
  • Elimination: We manipulate the equations so that when we add or subtract them, one of the variables cancels out. Again, this leaves us with a single equation in one variable.

Which method we choose often depends on the specific equations we're dealing with. Sometimes one method is clearly easier than the other. For our system, we'll explore both, just to give you a comprehensive view.

When tackling system of equations, keep in mind that there are three possible scenarios for solutions. A system can have:

  1. Exactly one solution: This means the lines intersect at a single point.
  2. No solution: This means the lines are parallel and never intersect.
  3. Infinitely many solutions: This means the lines are actually the same line, just written in different forms.

Okay, with the basics covered, let's roll up our sleeves and get to solving!

Method 1: The Substitution Method

Let's start by using the substitution method to solve our system. Remember, the system we're working with is:

  1. -3x + 4y = 12
  2. (1/4)x - (1/3)y = 1

Step 1: Solve one equation for one variable.

Looking at our equations, it might be easier to solve the second equation for x. It avoids dealing with negative coefficients right away. So, let's isolate x in the second equation: (1/4)x - (1/3)y = 1

First, add (1/3)y to both sides:

(1/4)x = 1 + (1/3)y

Now, multiply both sides by 4 to get x by itself:

x = 4 + (4/3)y

Great! We've solved the second equation for x. Now we have an expression for x in terms of y. This is a crucial step in the substitution method.

Step 2: Substitute the expression into the other equation.

Now, we'll substitute the expression we just found for x (which is 4 + (4/3)y) into the first equation: -3x + 4y = 12

Replace x with our expression:

-3(4 + (4/3)y) + 4y = 12

See what we did there? We've now created an equation that only has y as a variable! This is the power of substitution.

Step 3: Solve for the remaining variable.

Let's simplify and solve for y. First, distribute the -3:

-12 - 4y + 4y = 12

Notice anything interesting? The -4y and +4y cancel each other out! This leaves us with:

-12 = 12

Wait a minute… -12 does not equal 12! This is a contradiction. What does this mean for our system of equations?

Step 4: Interpret the result.

The fact that we arrived at a contradiction (-12 = 12) tells us that our system of equations has no solution. Remember, this means the lines represented by these equations are parallel and never intersect. So, using the substitution method, we've already figured out the nature of the solution – or, in this case, the lack of a solution.

Method 2: The Elimination Method

Okay, let's confirm our findings using the elimination method. This method is super handy when you can easily manipulate the equations to cancel out a variable.

Again, our system is:

  1. -3x + 4y = 12
  2. (1/4)x - (1/3)y = 1

Step 1: Manipulate the equations to make coefficients match (or be opposites).

To use elimination, we need to make the coefficients of either x or y the same (or opposites) in both equations. Let's target the x variable. To do this, we can multiply the second equation by 12. Why 12? Because 12 is the least common multiple of 4 (the denominator in (1/4)x) and the implicit denominator of 1 in the first equation's x term.

Multiply the second equation by 12:

12 * [(1/4)x - (1/3)y] = 12 * 1

This gives us:

3x - 4y = 12

Now, our system looks like this:

  1. -3x + 4y = 12
  2. 3x - 4y = 12

Notice that the x coefficients are now opposites (-3 and 3). This is perfect for elimination!

Step 2: Add or subtract the equations to eliminate a variable.

Since the x coefficients are opposites, we'll add the two equations together:

(-3x + 4y) + (3x - 4y) = 12 + 12

This simplifies to:

0 = 24

Step 3: Interpret the result.

Just like with the substitution method, we've arrived at a contradiction (0 = 24). This confirms our earlier finding: the system of equations has no solution.

Visualizing the Solution (or Lack Thereof)

To really drive home the concept, let's think about what this means graphically. Each of our equations represents a line. When a system has one solution, the lines intersect at a single point. When a system has infinitely many solutions, the lines are actually the same line (they overlap completely). But when a system has no solution, the lines are parallel. They have the same slope but different y-intercepts, so they never cross.

If we were to graph our equations, we'd see two parallel lines, visually confirming our algebraic solution.

Key Takeaways

  • Systems of equations can have one solution, no solution, or infinitely many solutions.
  • The substitution and elimination methods are powerful tools for solving systems of equations.
  • A contradiction (like -12 = 12 or 0 = 24) indicates that the system has no solution.
  • Graphically, a system with no solution represents parallel lines.

Conclusion

So, to wrap it up, guys, we've thoroughly explored the system of equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1. We used both the substitution and elimination methods and discovered that this system has no solution. Remember, this means the lines represented by these equations are parallel. Hopefully, this step-by-step guide has made solving systems of equations a little less mysterious and a lot more manageable. Keep practicing, and you'll become a system-solving pro in no time! Keep rocking it, Plastik Magazine fam!