Solving Systems Of Equations A Step By Step Guide

Jul 30, 2025 by James Vasile 50 views

Solve the System of Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of simultaneous equations and tackling a specific problem. We've got a system of two equations with two variables (x and y), and our mission is to find the values of x and y that satisfy both equations at the same time. It might sound a bit intimidating, but trust me, we'll break it down into easy-to-follow steps. Let's get started!

The Problem

Here's the system of equations we need to solve:

-3x + 6y = 9
5x + 7y = -49

And here are the possible solutions:

A. (1, -2) B. (-2, -7) C. (-7, -2) D. (-2, 1/2)

Understanding Systems of Equations

Before we jump into solving this specific system, let's take a moment to understand what a system of equations actually represents. Think of each equation as a line drawn on a graph. The solution to the system is the point where these lines intersect. In other words, it's the (x, y) coordinate that lies on both lines. There are several methods we can use to find this intersection point, including substitution, elimination, and graphing. For this problem, we'll focus on the elimination method, as it's often the most efficient for systems like this one. The core idea behind the elimination method is to manipulate the equations so that when we add them together, one of the variables cancels out. This leaves us with a single equation with a single variable, which we can easily solve. Then, we can substitute that value back into one of the original equations to find the value of the other variable. We need to manipulate the equations in such a way that the coefficients of either x or y are opposites (e.g., 3 and -3, or -6 and 6). This is typically done by multiplying one or both equations by a constant. The trick is to choose the right constants so that when we add the equations, one of the variables disappears. Once we have a value for one variable, we simply substitute it back into either of the original equations to solve for the other variable. This process of substitution transforms the two-variable problem into a single-variable problem, which is much easier to solve. The solution we obtain represents the point of intersection of the two lines represented by the equations. This point is the unique solution that satisfies both equations simultaneously. Understanding this graphical interpretation can be incredibly helpful in visualizing the problem and checking our answers. If we were to graph these equations, the solution we find algebraically would correspond to the coordinates of the point where the two lines intersect.

Solving by Elimination

Step 1: Multiply to Match Coefficients

Our goal is to make the coefficients of either x or y opposites. Looking at the equations, it seems easier to eliminate x. To do this, we can multiply the first equation by 5 and the second equation by 3:

(5) * (-3x + 6y) = (5) * 9  =>  -15x + 30y = 45
(3) * (5x + 7y) = (3) * -49 =>  15x + 21y = -147

Now, we have:

-15x + 30y = 45
15x + 21y = -147

Notice that the coefficients of x are now -15 and 15, which are opposites! This is exactly what we wanted.

Step 2: Add the Equations

Next, we add the two equations together:

(-15x + 30y) + (15x + 21y) = 45 + (-147)

Simplifying, we get:

51y = -102

See how the x terms canceled out? That's the power of the elimination method!

Step 3: Solve for y

Now we have a simple equation to solve for y. Divide both sides by 51:

y = -102 / 51
y = -2

Great! We've found the value of y. It's -2.

Step 4: Substitute to Find x

Now that we know y = -2, we can substitute this value into either of the original equations to solve for x. Let's use the first equation:

-3x + 6y = 9
-3x + 6(-2) = 9
-3x - 12 = 9

Add 12 to both sides:

-3x = 21

Divide both sides by -3:

x = -7

Awesome! We've found x = -7.

Step 5: The Solution

We've found that x = -7 and y = -2. This means the solution to the system of equations is the ordered pair (-7, -2).

Therefore, the correct answer is C. (-7, -2).

Checking Our Work

It's always a good idea to check our solution to make sure we didn't make any mistakes. We can do this by substituting the values of x and y back into both of the original equations.

Let's start with the first equation:

-3x + 6y = 9
-3(-7) + 6(-2) = 9
21 - 12 = 9
9 = 9  (This is true!)

Now, let's check the second equation:

5x + 7y = -49
5(-7) + 7(-2) = -49
-35 - 14 = -49
-49 = -49  (This is also true!)

Since our solution satisfies both equations, we can be confident that it's correct.

Key Takeaways

The elimination method is a powerful tool for solving systems of equations. It involves manipulating the equations to eliminate one variable, making it easier to solve for the other. Mastery of this method opens doors to solving complex mathematical problems efficiently. The beauty of this method lies in its systematic approach. By strategically multiplying equations, we create opportunities for variables to cancel out, simplifying the problem into manageable steps. This systematic nature makes it less prone to errors and provides a clear pathway to the solution.
Checking your work is crucial. Always substitute your solution back into the original equations to ensure accuracy. Developing this habit not only ensures correct answers but also strengthens understanding of the underlying concepts. This practice helps to identify any subtle errors in calculations and reinforces the relationship between the variables and the equations.
Systems of equations represent real-world relationships. They are used extensively in various fields like economics, engineering, and computer science. Learning to solve them is a valuable skill that has applications far beyond the classroom. From modeling supply and demand in economics to designing circuits in engineering, systems of equations provide a powerful framework for analyzing and solving real-world problems. Understanding these applications can provide a deeper appreciation for the importance of this mathematical concept.

Alternative Methods

While we used the elimination method here, it's worth noting that there are other methods to solve systems of equations. One common method is substitution, where you solve one equation for one variable and substitute that expression into the other equation. Another method is graphing, where you plot both equations as lines and find their intersection point. Each method has its strengths and weaknesses, and the best method to use often depends on the specific problem. For instance, the substitution method is particularly useful when one of the equations is already solved for one variable. The graphing method provides a visual representation of the solution and can be helpful for understanding the concept of simultaneous equations, although it may not always provide exact solutions, especially when the solutions are not integers. Exploring these different methods not only provides flexibility in problem-solving but also enhances a deeper understanding of the underlying mathematical principles.

Practice Makes Perfect

The best way to master solving systems of equations is to practice! Try solving more problems using the elimination method, substitution method, and graphing method. You'll develop a feel for which method works best in different situations, and you'll become more confident in your ability to solve these types of problems. Remember, mathematics is not a spectator sport. The more you actively engage with the problems, the more proficient you will become. Seek out additional practice problems in textbooks, online resources, or from your instructor. Don't hesitate to try different approaches and learn from both your successes and mistakes. The journey of learning mathematics is a process of continuous exploration and refinement. By embracing the challenges and actively engaging in practice, you will unlock the power of mathematical thinking and problem-solving. So, keep practicing, keep exploring, and keep challenging yourself to reach new heights in your mathematical journey!

Conclusion

So there you have it! We've successfully solved the system of equations using the elimination method. Remember the steps: match coefficients, add the equations, solve for one variable, substitute to find the other, and always check your work. With practice, you'll be solving systems of equations like a pro in no time!