Solving Systems Of Equations Using Tables 2x - 2y = 6 And 4x + 4y = 28

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In the realm of mathematics, solving systems of equations is a fundamental skill. This article delves into a specific problem: finding the solution to the system of equations $2x - 2y = 6$ and $4x + 4y = 28$. We will explore how a table can be used to systematically determine the solution, providing a step-by-step guide that is both comprehensive and easy to understand. Our discussion will not only focus on identifying the correct answer but also on understanding the underlying principles and techniques involved in solving such problems.

Understanding the Problem

The problem presents us with two linear equations:

  1. 2x−2y=62x - 2y = 6

  2. 4x+4y=284x + 4y = 28

Our goal is to find a pair of values $(x, y)$ that satisfies both equations simultaneously. This pair of values represents the point of intersection of the two lines represented by these equations. A common approach to solving such systems is to use methods like substitution, elimination, or graphical methods. However, the problem suggests using a table to find the solution, which implies a systematic trial-and-error approach. This method involves testing different pairs of values for $(x, y)$ and checking if they satisfy both equations. This approach is particularly useful when dealing with multiple-choice questions, as it allows us to quickly verify the given options.

The Table Method: A Step-by-Step Approach

The table method involves creating a table with columns for xx, yy, and the results of substituting these values into each equation. We then test the given options one by one until we find a pair that satisfies both equations. This method is straightforward and can be very effective, especially when the options provide a limited set of possibilities. Let's break down the process into detailed steps:

  1. Set up the table: Create a table with columns for xx, yy, $2x - 2y$, and $4x + 4y$. This will help you organize your calculations and keep track of the results.
  2. Test each option: For each option $(x, y)$, substitute the values of xx and yy into both equations.
  3. Check for equality: Compare the results of the substitutions with the constants on the right-hand side of the equations. If both equations are satisfied, then you have found the solution.
  4. Eliminate incorrect options: If an option does not satisfy both equations, eliminate it and move on to the next option.

By following these steps, we can systematically test each option and identify the correct solution. This method not only helps in finding the answer but also reinforces the understanding of how solutions to systems of equations work. Each option represents a potential point in the coordinate plane, and we are essentially checking which point lies on both lines defined by the equations.

Testing the Options

Now, let's apply the table method to the given options:

A. $(2, 5)$

  • Substitute $x = 2$ and $y = 5$ into the first equation: $2(2) - 2(5) = 4 - 10 = -6$. This does not equal 6, so option A is incorrect.

B. $(5, 2)$

  • Substitute $x = 5$ and $y = 2$ into the first equation: $2(5) - 2(2) = 10 - 4 = 6$. This satisfies the first equation.
  • Substitute $x = 5$ and $y = 2$ into the second equation: $4(5) + 4(2) = 20 + 8 = 28$. This satisfies the second equation.

Since $(5, 2)$ satisfies both equations, option B is the correct solution. We can stop here, but for the sake of completeness, let's test the other options as well.

C. $(5, -8)$

  • Substitute $x = 5$ and $y = -8$ into the first equation: $2(5) - 2(-8) = 10 + 16 = 26$. This does not equal 6, so option C is incorrect.

As we have found a solution in option B, there's no need to test further options. However, it's always a good practice to understand why other options are incorrect, which reinforces the understanding of the solution process. The key takeaway here is that a solution to a system of equations must satisfy all equations in the system. If a pair of values fails to satisfy even one equation, it cannot be the solution.

Why $(5, 2)$ is the Solution: A Deeper Dive

Option B, $(5, 2)$, is the correct solution because it makes both equations true simultaneously. Let's break down why this is the case and how this solution relates to the graphical representation of these equations.

  • Equation 1: $2x - 2y = 6$ When we substitute $x = 5$ and $y = 2$, we get $2(5) - 2(2) = 10 - 4 = 6$, which is true. This means the point $(5, 2)$ lies on the line represented by this equation.
  • Equation 2: $4x + 4y = 28$ Similarly, when we substitute $x = 5$ and $y = 2$, we get $4(5) + 4(2) = 20 + 8 = 28$, which is also true. This means the point $(5, 2)$ lies on the line represented by this equation as well.

Graphically, each linear equation represents a straight line. The solution to a system of two linear equations is the point where these two lines intersect. The point of intersection is the only point that lies on both lines, and therefore, it is the only pair of values $(x, y)$ that satisfies both equations. In this case, the lines represented by $2x - 2y = 6$ and $4x + 4y = 28$ intersect at the point $(5, 2)$.

Alternative Methods for Solving Systems of Equations

While the table method is effective for multiple-choice questions, it's important to be aware of other methods for solving systems of equations. These methods provide a more general approach and can be used even when options are not provided. Two common methods are:

  1. Substitution Method: In this method, we solve one equation for one variable in terms of the other variable, and then substitute this expression into the other equation. This reduces the system to a single equation with one variable, which can be easily solved. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.
  2. Elimination Method: In this method, we manipulate the equations so that the coefficients of one of the variables are opposites. Then, we add the equations together, which eliminates one variable and leaves us with a single equation with one variable. Again, we can solve for one variable and then substitute back to find the other.

For this particular problem, let's demonstrate the elimination method:

  1. Multiply the first equation by 2: This gives us $4x - 4y = 12$.
  2. Add the modified first equation to the second equation: $(4x - 4y) + (4x + 4y) = 12 + 28$, which simplifies to $8x = 40$.
  3. Solve for x: $x = rac{40}{8} = 5$.
  4. Substitute x = 5 into either original equation: Let's use the first equation: $2(5) - 2y = 6$, which simplifies to $10 - 2y = 6$.
  5. Solve for y: $-2y = -4$, so $y = 2$.

Thus, the solution is $(x, y) = (5, 2)$, which confirms our previous result using the table method. This demonstrates the power of alternative methods and how they can be used to verify solutions found using other techniques.

Common Mistakes and How to Avoid Them

When solving systems of equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy.

  1. Arithmetic Errors: One of the most common mistakes is making arithmetic errors during the substitution or elimination process. This can lead to incorrect solutions. To avoid this, it's crucial to double-check your calculations at each step.
  2. Incorrect Substitution: When using the substitution method, it's important to substitute the expression correctly into the other equation. Substituting into the same equation you solved for can lead to a circular process and no solution.
  3. Sign Errors: Sign errors are particularly common when dealing with negative numbers. Pay close attention to the signs when adding, subtracting, multiplying, or dividing. For example, when distributing a negative sign, make sure to change the signs of all terms inside the parentheses.
  4. Forgetting to Solve for Both Variables: A solution to a system of equations consists of values for all variables. Don't stop after finding the value of one variable; make sure to substitute back to find the value of the other variable(s).
  5. Not Checking the Solution: Always check your solution by substituting the values back into the original equations. This is a quick way to catch errors and ensure that your solution is correct. If the solution does not satisfy both equations, there is a mistake in your calculations.

By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in solving systems of equations. Practice is key, so work through a variety of problems to solidify your understanding of the different methods and techniques.

Conclusion: Mastering Systems of Equations

In conclusion, solving systems of equations is a crucial skill in mathematics, and understanding different methods for finding solutions is essential. In this article, we have explored how to use a table to determine the solution of the equations $2x - 2y = 6$ and $4x + 4y = 28$. We found that the solution is $(5, 2)$, which satisfies both equations. We also discussed alternative methods such as substitution and elimination, and highlighted common mistakes to avoid. By mastering these techniques and practicing regularly, you can confidently tackle a wide range of problems involving systems of equations. Remember, the key to success in mathematics is not just memorizing formulas, but also understanding the underlying concepts and developing problem-solving strategies. Whether you prefer the systematic approach of the table method or the algebraic elegance of substitution and elimination, the ability to solve systems of equations is a valuable asset in your mathematical toolkit.