Solve System Of Equations: 3x + 4y = 10 And 6x - 2y = 40

by Alex Johnson 57 views

Are you struggling with systems of equations? Don't worry, you're not alone! Many students find these problems tricky, but with the right approach, they become much easier to handle. In this article, we'll break down a specific example and walk you through the process of finding the solution. We'll focus on the system of equations:

3x + 4y = 10
6x - 2y = 40

We need to determine which of the following options is the correct solution:

A. (-2, 6) B. (2, 0) C. (6, 2) D. (2, 6)

Let’s dive in and find the correct answer together!

Understanding Systems of Equations

Before we jump into solving the problem, let's quickly recap what a system of equations actually is. A system of equations is simply a set of two or more equations that involve the same variables. The goal is to find values for these variables that satisfy all equations in the system simultaneously. In our case, we have two equations with two variables, x and y. The solution will be a pair of values (x, y) that make both equations true.

There are several methods to solve systems of equations, including:

  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination (or Addition/Subtraction): Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations together to eliminate that variable.
  • Graphing: Graph both equations and find the point of intersection, which represents the solution.

For this particular problem, the elimination method seems like a straightforward approach. Let's see how it works.

Applying the Elimination Method

The elimination method involves manipulating the equations so that when you add them together, one of the variables cancels out. Looking at our equations:

3x + 4y = 10
6x - 2y = 40

We can see that if we multiply the first equation by -2, the coefficient of x will become -6, which is the opposite of the coefficient of x in the second equation (6). This will allow us to eliminate x when we add the equations.

So, let’s multiply the first equation by -2:

-2 * (3x + 4y) = -2 * 10
-6x - 8y = -20

Now we have a modified system:

-6x - 8y = -20
6x - 2y = 40

Next, we add the two equations together. Notice how the x terms cancel out:

(-6x - 8y) + (6x - 2y) = -20 + 40
-10y = 20

Now we can solve for y:

y = 20 / -10
y = -2

So, we've found that y = -2. Now we need to find the value of x. We can substitute this value of y into either of the original equations. Let's use the first equation, 3x + 4y = 10:

3x + 4(-2) = 10
3x - 8 = 10

Add 8 to both sides:

3x = 18

Divide by 3:

x = 6

Therefore, the solution to the system of equations is x = 6 and y = -2. This corresponds to the ordered pair (6, -2).

Checking the Solution

It’s always a good idea to check your solution to make sure it’s correct. We can do this by plugging the values of x and y back into both original equations.

For the first equation, 3x + 4y = 10:

3(6) + 4(-2) = 18 - 8 = 10

This equation holds true.

For the second equation, 6x - 2y = 40:

6(6) - 2(-2) = 36 + 4 = 40

This equation also holds true. Since our solution (6, -2) satisfies both equations, we can be confident that it is the correct answer.

Identifying the Correct Option

Looking back at the options:

A. (-2, 6) B. (2, 0) C. (6, 2) D. (2, 6)

None of these options match our solution of (6, -2). There seems to be a mistake in the provided options. The correct solution is (6, -2), which is not listed. It's important to recognize when an answer isn't present and double-check your work, or in this case, consider that the provided choices might be incorrect.

Common Mistakes and How to Avoid Them

Solving systems of equations can be prone to errors if you're not careful. Here are some common mistakes and how to avoid them:

  1. Arithmetic Errors: Simple addition, subtraction, multiplication, or division mistakes can throw off the entire solution. Always double-check your calculations! Use a calculator if necessary, especially for larger numbers.
  2. Sign Errors: Pay close attention to the signs (positive and negative) of the terms. A single sign error can lead to an incorrect solution. Write down each step clearly and double-check the signs as you go.
  3. Incorrectly Applying the Elimination Method: Make sure you multiply the entire equation by the constant, not just one term. Distribute the constant properly to all terms in the equation. Also, ensure that the coefficients you are trying to eliminate are opposites before adding the equations.
  4. Incorrectly Applying the Substitution Method: When substituting, make sure you substitute the expression into the correct equation and that you simplify the resulting equation correctly. Double-check which equation you are substituting into and be careful with the order of operations.
  5. Forgetting to Solve for Both Variables: Remember that the solution to a system of equations is a pair of values (x, y). Don't stop after finding one variable; you need to find the value of both variables.
  6. Not Checking the Solution: As we demonstrated, checking your solution is crucial to ensure accuracy. Always plug your solution back into the original equations to verify that they hold true.

Alternative Methods for Solving Systems of Equations

While we used the elimination method in this example, it's worth briefly mentioning other methods you can use to solve systems of equations.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. For example, consider the system:

x + y = 5
2x - y = 1

We can solve the first equation for x:

x = 5 - y

Now, substitute this expression for x into the second equation:

2(5 - y) - y = 1
10 - 2y - y = 1
10 - 3y = 1
-3y = -9
y = 3

Now substitute y = 3 back into the equation x = 5 - y:

x = 5 - 3
x = 2

So the solution is (2, 3).

Graphing Method

The graphing method involves plotting both equations on a coordinate plane. The point where the lines intersect represents the solution to the system. This method is particularly useful for visualizing the solutions, but it may not be the most accurate method if the solutions are not integers.

Conclusion

Solving systems of equations is a fundamental skill in algebra. By understanding the different methods available (elimination, substitution, and graphing) and practicing regularly, you can master these problems. Remember to double-check your work and be aware of common mistakes. In the example we worked through, we found the solution to be (6, -2), but this option wasn't provided in the given choices, highlighting the importance of verifying your answer and recognizing potential errors in the problem itself. Keep practicing, and you'll become a pro at solving systems of equations!

For further learning and practice, you might find helpful resources at Khan Academy Algebra.