Solving Systems Of Linear Equations A Comprehensive Guide

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Hey guys! Let's dive into the fascinating world of systems of linear equations (SPLDV). If you've ever encountered two equations with two unknowns (usually x and y), you're in the right place. We're going to break down how to solve these equations using not one, but four different methods. Yep, you heard that right! We'll cover graphing, substitution, elimination, and a combination method. So, buckle up, grab a pen and paper, and let's get started!

The Problem at Hand

Before we jump into the methods, let's define the problem we're going to solve. We have two equations:

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

Our mission, should we choose to accept it (and we definitely do!), is to find the values of x and y that satisfy both equations simultaneously. These values will be the solution to our system of equations.

1. The Graphical Method: Visualizing the Solution

The graphical method is a fantastic way to visualize what we're actually doing when solving a system of equations. Think of each equation as representing a straight line on a graph. The solution to the system is the point where these two lines intersect. Cool, right?

Step-by-Step Guide

  1. Rewrite the equations in slope-intercept form (y = mx + b): This form makes it super easy to graph the lines.

    • Equation 1: 2x - 3y = -12
      • Subtract 2x from both sides: -3y = -2x - 12
      • Divide both sides by -3: y = (2/3)x + 4
    • Equation 2: x + 2y = 4
      • Subtract x from both sides: 2y = -x + 4
      • Divide both sides by 2: y = (-1/2)x + 2

  2. Identify the slope (m) and y-intercept (b) for each equation:

    • Equation 1: y = (2/3)x + 4
      • Slope (m) = 2/3
      • Y-intercept (b) = 4
    • Equation 2: y = (-1/2)x + 2
      • Slope (m) = -1/2
      • Y-intercept (b) = 2

  3. Plot the lines on a graph:

    • For each line, start by plotting the y-intercept. Then, use the slope to find another point. Remember, slope (m) = rise/run. So, if the slope is 2/3, you go up 2 units and right 3 units from the y-intercept.
    • Draw a straight line through the two points for each equation.

  4. Identify the point of intersection: This is where the magic happens! The coordinates of the point where the two lines cross are the solution to the system of equations.

Finding the Intersection

Looking at the graph, we can see that the lines intersect at the point (-2, 8/3). Therefore, x = -2 and y = 8/3 is our graphical solution.

The graphical method is awesome because it gives you a visual representation of the solution. You can literally see the point where the equations meet. However, it can be less precise if the intersection point doesn't fall perfectly on grid lines. That's where our other methods come in handy.

2. The Substitution Method: A Step-by-Step Approach

The substitution method is all about isolating one variable in one equation and then substituting that expression into the other equation. This allows us to solve for one variable, and then we can easily find the other.

Step-by-Step Guide

  1. Solve one equation for one variable: Choose the equation and variable that looks easiest to isolate. In our case, Equation 2 (x + 2y = 4) seems like a good candidate. Let's solve it for x:

    • x + 2y = 4
    • Subtract 2y from both sides: x = 4 - 2y

  2. Substitute the expression into the other equation: Now, we'll take the expression we found for x (4 - 2y) and substitute it into Equation 1 (2x - 3y = -12):

    • 2(4 - 2y) - 3y = -12

  3. Solve for the remaining variable: We now have an equation with only one variable (y). Let's solve for it:

    • 8 - 4y - 3y = -12
    • Combine like terms: 8 - 7y = -12
    • Subtract 8 from both sides: -7y = -20
    • Divide both sides by -7: y = 20/7

  4. Substitute back to find the other variable: Now that we know y = 20/7, we can plug it back into either Equation 1 or Equation 2 to solve for x. It's often easiest to use the equation where you've already isolated a variable. So, let's use x = 4 - 2y:

    • x = 4 - 2(20/7)
    • x = 4 - 40/7
    • x = 28/7 - 40/7
    • x = -12/7

The Solution

Using the substitution method, we found that x = -12/7 and y = 20/7. This method is powerful because it allows us to systematically eliminate one variable at a time.

3. The Elimination Method: The Art of Cancellation

The elimination method, sometimes called the addition method, focuses on eliminating one variable by adding or subtracting the equations. The key is to manipulate the equations so that the coefficients of one variable are opposites.

Step-by-Step Guide

  1. Multiply one or both equations to make the coefficients of one variable opposites: Looking at our equations:

    • 2x - 3y = -12
    • x + 2y = 4

    We can eliminate x by multiplying the second equation by -2:

    • -2(x + 2y) = -2(4)
    • -2x - 4y = -8

    Now we have:

    • 2x - 3y = -12
    • -2x - 4y = -8

  2. Add the equations together: Now, we add the two equations vertically:

    • (2x - 3y) + (-2x - 4y) = -12 + (-8)
    • -7y = -20

  3. Solve for the remaining variable:

    • Divide both sides by -7: y = 20/7

  4. Substitute back to find the other variable: Just like with the substitution method, we plug y = 20/7 back into either of the original equations to solve for x. Let's use x + 2y = 4:

    • x + 2(20/7) = 4
    • x + 40/7 = 4
    • x = 4 - 40/7
    • x = 28/7 - 40/7
    • x = -12/7

Elimination in Action

The elimination method gave us the solution x = -12/7 and y = 20/7. This method is especially useful when the equations are already set up in a way that makes it easy to eliminate a variable.

4. The Combination Method: Best of Both Worlds

The combination method, as the name suggests, combines the power of both substitution and elimination. This approach can be particularly helpful when dealing with more complex systems of equations.

Step-by-Step Guide

  1. Use elimination to simplify the system: Let's start by eliminating x, just like we did in the elimination method. We'll multiply the second equation by -2:

    • -2(x + 2y) = -2(4)
    • -2x - 4y = -8

    Now we have:

    • 2x - 3y = -12
    • -2x - 4y = -8

    Add the equations together:

    • (2x - 3y) + (-2x - 4y) = -12 + (-8)
    • -7y = -20

    Solve for y:

    • y = 20/7

  2. Use substitution to find the other variable: Now that we have y = 20/7, we can substitute it back into either of the original equations to solve for x. Let's use x + 2y = 4:

    • x + 2(20/7) = 4
    • x + 40/7 = 4
    • x = 4 - 40/7
    • x = 28/7 - 40/7
    • x = -12/7

Combining for Success

The combination method nicely confirms our previous results: x = -12/7 and y = 20/7. This method allows you to choose the most efficient approach for each step of the solution.

Conclusion: Four Methods, One Solution

We've successfully navigated the world of systems of linear equations using four different methods: graphical, substitution, elimination, and combination. Each method has its strengths and weaknesses, but they all lead to the same solution. The values x = -12/7 and y = 20/7 satisfy both equations in our system.

So, next time you encounter an SPLDV, you'll be well-equipped to tackle it head-on. Whether you prefer the visual approach of graphing, the systematic substitution, the elegant elimination, or the flexible combination method, you've got the tools to conquer those equations. Keep practicing, and you'll become a system-solving pro in no time! Remember guys, math can be fun, especially when you have multiple ways to solve a problem.