Solve Equations: Linear Combination Method Explained

by Henrik Larsen 53 views

Hey guys! Let's dive into a fun and super useful method in mathematics: the linear combination method for solving systems of equations. This technique is a powerful tool for finding solutions where two or more equations are working together. In this article, we'll break down how the linear combination method works, walk through a specific example step-by-step, and highlight why it's such an important skill to have in your math arsenal. So, buckle up, and let's get started!

Understanding the Linear Combination Method

The linear combination method, also known as the elimination method, is a clever way to solve systems of linear equations. Imagine you have two equations with two variables, like x and y. The goal is to manipulate these equations in such a way that when you add or subtract them, one of the variables disappears, leaving you with a single equation in one variable. This makes it much easier to solve for the remaining variable. Once you find the value of one variable, you can plug it back into either of the original equations to find the value of the other variable. Think of it as a mathematical magic trick where we make one variable vanish to reveal the other!

The Key Steps

So, how do we perform this magic trick? Here are the key steps involved in the linear combination method:

  1. Align the Equations: First, make sure your equations are lined up nicely, with the x terms, y terms, and constants each in their own columns. This will make it easier to see what to do next.
  2. Multiply (if needed): Look at the coefficients (the numbers in front of the variables). If necessary, multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 2 and -2) or the same (e.g., 3 and 3). This is the crucial step that sets up the elimination.
  3. Add or Subtract: Now, add or subtract the equations. If the coefficients were opposites, adding will eliminate the variable. If they were the same, subtracting will do the trick. You should now have a single equation with one variable.
  4. Solve: Solve the resulting equation for the remaining variable. This is usually a straightforward algebraic step.
  5. Substitute: Take the value you just found and substitute it back into either of the original equations. This will give you an equation with only one variable left, which you can solve for the other variable.
  6. Check: Finally, check your solution by plugging both values back into both original equations to make sure they hold true. This is a great way to catch any mistakes.

Why Use Linear Combinations?

You might be wondering, why bother with all these steps? Why not just use another method, like substitution? Well, the linear combination method is particularly useful when the equations are already in a form that makes it easy to eliminate a variable. It can be more efficient than substitution in many cases, especially when dealing with larger systems of equations (more than two variables). Plus, mastering this method gives you another powerful tool in your problem-solving toolbox, which is always a good thing!

Example: Walking Through a Solution

Okay, let's put the linear combination method into action with a specific example. This will make the process even clearer. We'll use the example provided and break down each step.

The original system of equations is presented with some initial manipulations:

4(.25x+5y=3.75)→x+20y=1514(4x−8y=12)→x−2y=3\begin{array}{r} 4(.25 x+5 y=3.75) \rightarrow x+20y=15 \\ \frac{1}{4}(4 x-8 y=12) \rightarrow x-2 y=3 \\ \end{array}

The goal here is to find the values of x and y that satisfy both equations.

Step-by-Step Solution

Let's analyze the given steps and complete the solution.

  1. Simplify the Equations:

    The initial step involves simplifying both equations:

    • Equation 1: 4(.25x + 5y = 3.75) simplifies to x + 20y = 15
    • Equation 2: 1/4(4x - 8y = 12) simplifies to x - 2y = 3

    These simplifications make the equations easier to work with. We've distributed the constants outside the parentheses to get rid of them.

  2. Align and Prepare for Elimination:

    Now we have the simplified system:

    • x + 20y = 15
    • x - 2y = 3

    Notice that the coefficients of x in both equations are the same (both are 1). This means we can easily eliminate x by subtracting one equation from the other.

  3. Eliminate x:

    Subtract the second equation from the first:

    (x + 20y) - (x - 2y) = 15 - 3

    This simplifies to:

    22y = 12

    See how the x terms canceled out? That's the magic of linear combinations!

  4. Solve for y:

    Now we have a simple equation to solve for y:

    22y = 12

    Divide both sides by 22:

    y = 12/22 = 6/11

    So, we've found the value of y!

  5. Substitute to Find x:

    Now that we know y = 6/11, we can substitute this value back into either of the simplified equations to solve for x. Let's use the second equation (x - 2y = 3) because it looks a bit simpler:

    x - 2(6/11) = 3

    x - 12/11 = 3

    Add 12/11 to both sides:

    x = 3 + 12/11

    x = 33/11 + 12/11

    x = 45/11

    So, we've found the value of x!

  6. The Solution:

    Therefore, the solution to the system of equations is:

    x = 45/11 and y = 6/11

Checking Our Work

It's always a good idea to check our solution. Let's plug these values back into the original simplified equations:

  • Equation 1: x + 20y = 15

    (45/11) + 20(6/11) = 45/11 + 120/11 = 165/11 = 15 (Correct!)

  • Equation 2: x - 2y = 3

    (45/11) - 2(6/11) = 45/11 - 12/11 = 33/11 = 3 (Correct!)

Since our solution satisfies both equations, we know we've done it right!

Why This Matters: Real-World Applications

The linear combination method isn't just an abstract math concept. It has tons of real-world applications. Systems of equations pop up all over the place, from science and engineering to economics and computer science. For example:

  • Mixing Solutions: Imagine you're a chemist mixing two solutions with different concentrations to get a desired concentration. You can use a system of equations to figure out how much of each solution to use.
  • Supply and Demand: Economists use systems of equations to model the relationship between supply and demand in a market. The solution tells them the equilibrium price and quantity.
  • Circuit Analysis: Electrical engineers use systems of equations to analyze circuits and determine the current and voltage in different parts of the circuit.
  • Computer Graphics: Computer graphics programmers use systems of equations to perform transformations on objects in 3D space, like rotations and scaling.

These are just a few examples, but they illustrate how the ability to solve systems of equations is a valuable skill in many fields.

Conclusion: Mastering the Technique

The linear combination method is a powerful and versatile technique for solving systems of equations. By understanding the steps involved and practicing with examples, you can master this method and add another valuable tool to your mathematical toolkit. Remember, the key is to align the equations, manipulate them to eliminate a variable, solve for the remaining variable, and then substitute back to find the other variable. And always check your work! So, keep practicing, and you'll become a pro at solving systems of equations in no time. You got this!