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

Are you struggling with systems of equations? Don't worry, you're not alone! Many students find these problems tricky, but with a clear method, they become much easier to handle. In this article, we'll walk through how to solve a specific system of equations, step by step. Let's dive in!

Understanding Systems of Equations

Before we jump into the solution, let’s make sure we understand what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations in the system simultaneously. In other words, we're looking for the point where the lines represented by these equations intersect on a graph.

Why are systems of equations important? They're used to model real-world scenarios involving multiple variables and constraints, from balancing chemical equations to optimizing business decisions. Mastering the techniques to solve these systems is a crucial skill in mathematics and its applications.

There are several methods for solving systems of equations, including graphing, substitution, and elimination. We'll focus on the elimination method here, as it's particularly efficient for the given problem.

The Given System of Equations

We're given the following system of equations:

-3x + 5y = -2
3x + 7y = 26

Our task is to find the values of x and y that make both of these equations true. Let's break down the elimination method and apply it to this system.

Step-by-Step Solution Using the Elimination Method

The elimination method involves manipulating the equations so that when you add them together, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve.

Step 1: Notice the Coefficients

Look at the coefficients of x in both equations. We have -3x in the first equation and 3x in the second equation. Notice that these coefficients are opposites of each other. This is perfect for the elimination method!

Step 2: Add the Equations

Since the x coefficients are opposites, we can simply add the two equations together. This will eliminate the x variable:

(-3x + 5y) + (3x + 7y) = -2 + 26

Step 3: Simplify the Result

Combine like terms:

-3x + 3x + 5y + 7y = 24
0x + 12y = 24
12y = 24

Step 4: Solve for y

Divide both sides of the equation by 12 to isolate y:

12y / 12 = 24 / 12
y = 2

So, we've found that y = 2. Great! Now we need to find the value of x.

Step 5: Substitute y into One of the Original Equations

Choose either of the original equations. Let's use the first one:

-3x + 5y = -2

Substitute y = 2 into the equation:

-3x + 5(2) = -2

Step 6: Simplify and Solve for x

Simplify the equation:

-3x + 10 = -2

Subtract 10 from both sides:

-3x = -12

Divide both sides by -3:

x = 4

So, we've found that x = 4.

Step 7: Write the Solution as an Ordered Pair

The solution to the system of equations is the ordered pair (x, y), which is (4, 2). This means that the point (4, 2) is the intersection of the two lines represented by the equations.

Verifying the Solution

It's always a good idea to check your solution by plugging the values of x and y back into both original equations to make sure they hold true.

Check in the First Equation:

-3x + 5y = -2
-3(4) + 5(2) = -2
-12 + 10 = -2
-2 = -2  (True)

Check in the Second Equation:

3x + 7y = 26
3(4) + 7(2) = 26
12 + 14 = 26
26 = 26  (True)

Since the solution (4, 2) satisfies both equations, we can be confident that it's correct.

Analyzing Other Solution Possibilities

It's important to understand that not all systems of equations have a unique solution like the one we just found. There are a few possibilities:

1. Unique Solution: The lines intersect at one point, giving a single solution (like our example).
2. No Solution: The lines are parallel and never intersect. In this case, you'll get a contradiction when trying to solve the system (e.g., 0 = 5).
3. Infinitely Many Solutions: The lines are the same, meaning they overlap completely. Any point on the line is a solution. You'll get an identity when trying to solve the system (e.g., 0 = 0).

In our case, we found a unique solution, which means the lines intersect at exactly one point.

Conclusion

We've successfully solved the system of equations using the elimination method and found that the solution is (4, 2). Remember, practice is key to mastering these techniques. Keep working through different problems, and you'll become more confident in your ability to solve systems of equations. Understanding the underlying concepts and various solution methods will help you tackle more complex problems in mathematics and related fields.

If you're interested in learning more about systems of equations and other algebraic concepts, consider exploring resources like Khan Academy's Algebra section. They offer excellent explanations, practice problems, and videos to help you deepen your understanding.