Solving A System Of Equations: A Step-by-Step Guide
Let's dive into the world of algebra and tackle a system of equations! In this article, we'll break down the process of solving the system:
x - y = 4
3x - 3y = 12
We'll explore different methods and provide a clear, step-by-step guide to help you understand the solution. Whether you're a student brushing up on your skills or just curious about math, this guide is for you.
Understanding Systems of Equations
Before we jump into solving, let's quickly define what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all equations in the system true simultaneously. Think of it as finding the point where the lines represented by the equations intersect on a graph.
In our case, we have two equations with two variables (x and y). There are several ways to solve such a system, including substitution, elimination, and graphing. We'll focus on the elimination and substitution methods to illustrate the solution.
Method 1: Elimination Method
The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation with one variable, which you can easily solve. Let's apply this to our system:
x - y = 4 (Equation 1)
3x - 3y = 12 (Equation 2)
Notice that the coefficients of x and y in the second equation are multiples of the coefficients in the first equation. This is a clue that the elimination method might work well here. To make the coefficients of x the same, we can multiply Equation 1 by 3:
3 * (x - y) = 3 * 4 3x - 3y = 12 (Equation 3)
Now, observe that Equation 2 and Equation 3 are identical. This tells us something important: the two equations in the original system are actually representing the same line! When we have dependent equations, it means there are infinitely many solutions. Any pair of (x, y) values that satisfy one equation will also satisfy the other.
To further illustrate this, let’s subtract Equation 2 from Equation 3:
(3x - 3y) - (3x - 3y) = 12 - 12 0 = 0
This true statement confirms that the equations are dependent.
Infinite Solutions Explained
The fact that we have infinitely many solutions might seem confusing at first. Think of it this way: both equations describe the same line. Any point on that line is a solution to both equations. Imagine plotting these equations on a graph; you'd see only one line instead of two intersecting lines.
To express these infinite solutions, we can write the solution in terms of one variable. Let's solve Equation 1 for x:
x - y = 4 x = y + 4
This tells us that for any value of y, we can find a corresponding x value by adding 4 to y. The solution set can be written as {(x, y) | x = y + 4}, meaning “the set of all points (x, y) such that x equals y plus 4.”
Method 2: Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This also leads to a single equation with one variable. Let's see how it works with our system.
Starting with the same equations:
x - y = 4 (Equation 1)
3x - 3y = 12 (Equation 2)
Let's solve Equation 1 for x, as we did before:
x = y + 4
Now, substitute this expression for x into Equation 2:
3(y + 4) - 3y = 12
Distribute the 3:
3y + 12 - 3y = 12
Simplify:
12 = 12
Again, we arrive at a true statement. The variable y disappeared, which signifies that the equations are dependent and there are infinitely many solutions. The substitution method, just like the elimination method, leads us to the same conclusion.
Expressing the Solution Set with Substitution
As before, to express the solution set, we use the relationship we found between x and y: x = y + 4. The solution set remains {(x, y) | x = y + 4}.
This means that you can choose any value for y and then calculate x using the formula x = y + 4. For example:
- If y = 0, then x = 0 + 4 = 4. So, (4, 0) is a solution.
- If y = 1, then x = 1 + 4 = 5. So, (5, 1) is a solution.
- If y = -2, then x = -2 + 4 = 2. So, (2, -2) is a solution.
And so on. There are endless possibilities!
Why Infinite Solutions?
The key takeaway here is that the two equations represent the same line. Graphically, this means that if you were to plot both equations, you would only see one line. Any point on that line satisfies both equations, hence the infinite solutions.
In contrast, if the equations represented two distinct intersecting lines, there would be one unique solution (the point of intersection). If the equations represented parallel lines, there would be no solution, as the lines never intersect.
Recognizing dependent systems is an important skill in algebra. It allows you to understand the nature of the solutions and express them accurately.
Conclusion
In conclusion, the system of equations
x - y = 4
3x - 3y = 12
has infinitely many solutions. This is because the two equations are dependent, meaning they represent the same line. We can express the solution set as {(x, y) | x = y + 4}, indicating that for any value of y, we can find a corresponding x value that satisfies both equations. We arrived at this conclusion using both the elimination and substitution methods, demonstrating the versatility of these algebraic techniques.
Understanding systems of equations is fundamental to many areas of mathematics and its applications. By mastering these methods, you'll be well-equipped to tackle more complex problems in the future.
For further exploration of systems of equations and related topics, visit Khan Academy's Algebra I section. It's a great resource for learning and practicing algebra concepts.
