Subtracting Equations Solving A System Of Equations

JU07/09/2025 08, 2025 by THE IDEN 52 views

Solving Systems of Equations Subtraction Method

Are you grappling with solving systems of equations? Do you want to master the subtraction method? This comprehensive guide will walk you through a step-by-step solution to the problem: Subtract the second equation from the first in the following system:

-4x - 2y = -2
x - 2y = 9

By the end of this article, you'll not only know the answer but also understand the underlying principles of this vital algebraic technique. Let's dive in!

Understanding Systems of Equations

Before we delve into the solution, let's lay a solid foundation by understanding what systems of equations are and why they matter. A system of equations is a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. These systems appear in various fields, from engineering and physics to economics and computer science, making it a crucial concept to grasp. The subtraction method is a powerful tool for solving such systems.

The subtraction method, also known as the elimination method, is particularly useful when one or more variables have the same coefficient (or coefficients that are easy to make the same) in the equations. This allows us to eliminate one variable by subtracting one equation from the other, leaving us with a simpler equation in one variable that we can easily solve. It's a technique that simplifies complex problems, making them more manageable and straightforward. Understanding when and how to use this method effectively can significantly enhance your problem-solving skills in algebra and beyond.

Step-by-Step Solution

Let's tackle the problem head-on. We're given the following system of equations:

-4x - 2y = -2
x - 2y = 9

Our objective is to subtract the second equation from the first. This involves subtracting the left-hand side of the second equation from the left-hand side of the first equation, and similarly, subtracting the right-hand side of the second equation from the right-hand side of the first equation. This careful subtraction is the core of the method, allowing us to eliminate one variable and simplify the system.

Step 1: Perform the Subtraction

Subtract the second equation from the first:

(-4x - 2y) - (x - 2y) = -2 - 9

Carefully distribute the negative sign in the subtraction. This is a crucial step to avoid common errors. Distributing the negative sign correctly ensures that we account for the change in sign of each term in the second equation, which is essential for the subsequent simplification.

-4x - 2y - x + 2y = -11

Step 2: Simplify the Resulting Equation

Combine like terms:

(-4x - x) + (-2y + 2y) = -11
-5x = -11

Notice how the y terms cancel each other out. This is the magic of the subtraction method – eliminating one variable to make the equation simpler. By eliminating y, we've reduced the system to a single equation in one variable, which is much easier to solve. This simplification is the key to unlocking the solution.

Step 3: Solve for x

Divide both sides by -5 to isolate x:

x = -11 / -5
x = 11/5

We've now found the value of x. This is a significant step forward, as we've determined one of the variables in the solution. Knowing the value of x allows us to proceed to find the value of y, completing the solution to the system of equations.

Step 4: Substitute the Value of x into One of the Original Equations

Substitute x = 11/5 into the second equation:

(11/5) - 2y = 9

We choose the second equation for simplicity, but either equation would work. Substituting the value of x into one of the original equations is a crucial step to find the value of y. This substitution allows us to create an equation with only y as the unknown, making it solvable.

Step 5: Solve for y

Isolate y:

-2y = 9 - (11/5)
-2y = (45/5) - (11/5)
-2y = 34/5
y = (34/5) / -2
y = -17/5

We've successfully found the value of y. With both x and y values determined, we have the complete solution to the system of equations. This final step confirms our solution, providing the values that satisfy both original equations.

The Solution

The solution to the system of equations is:

x = 11/5
y = -17/5

Therefore, the result of subtracting the second equation from the first is x = 11/5 and y = -17/5. This solution represents the point where the two lines represented by the equations intersect on a graph. It's the unique pair of values that satisfies both equations simultaneously, making it the solution to the system.

Verification

To ensure our solution is correct, let's substitute these values back into both original equations:

Equation 1:

-4x - 2y = -2
-4(11/5) - 2(-17/5) = -2
(-44/5) + (34/5) = -2
-10/5 = -2
-2 = -2

Equation 2:

x - 2y = 9
(11/5) - 2(-17/5) = 9
(11/5) + (34/5) = 9
45/5 = 9
9 = 9

Both equations hold true, so our solution is correct! This verification step is essential to confirm the accuracy of the solution. By substituting the values back into the original equations and verifying that they hold true, we can be confident in our answer.

Common Mistakes to Avoid

When using the subtraction method, there are several common pitfalls to watch out for:

Incorrectly Distributing the Negative Sign: This is the most frequent error. Remember to distribute the negative sign to all terms in the equation being subtracted.
Combining Unlike Terms: Only combine terms with the same variable. For example, x terms can only be combined with other x terms.
Arithmetic Errors: Simple mistakes in addition, subtraction, multiplication, or division can lead to incorrect results. Double-check your calculations.
Forgetting to Substitute Back: After finding the value of one variable, remember to substitute it back into one of the original equations to find the value of the other variable.
Not Verifying the Solution: Always check your solution by substituting the values back into the original equations. This will catch any errors you may have made.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy when using the subtraction method.

Conclusion

Congratulations! You've successfully solved a system of equations using the subtraction method. We've walked through each step, from understanding the problem to verifying the solution. Remember, practice makes perfect. The more you work with systems of equations, the more comfortable and confident you'll become in solving them. Keep practicing, and you'll master this essential algebraic technique in no time!

The subtraction method is a valuable tool in your mathematical arsenal. It simplifies the process of solving systems of equations, making complex problems more manageable. By understanding the principles and practicing the steps, you can confidently tackle various mathematical challenges. Remember, math is a journey, and every problem solved is a step forward. Keep exploring, keep learning, and keep growing your mathematical skills!