Find An Ordered Pair { (x, Y)$}$ That Is A Solution To The Equation.${x - 6y = 6}${ (x, Y) = \square \square\$}

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Introduction

In mathematics, solving linear equations is a fundamental concept that involves finding the values of variables that satisfy a given equation. In this article, we will focus on finding an ordered pair solution to the equation x−6y=6x - 6y = 6. This equation is a linear equation in two variables, and we will use algebraic methods to solve for the values of xx and yy.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable(s) is 1. In the case of the equation x−6y=6x - 6y = 6, the highest power of the variables is 1, and the equation is linear in two variables, xx and yy. Linear equations can be solved using various methods, including substitution, elimination, and graphing.

The Equation x−6y=6x - 6y = 6

The equation x−6y=6x - 6y = 6 is a linear equation in two variables, xx and yy. To solve for the values of xx and yy, we can use the method of substitution or elimination. In this case, we will use the method of substitution.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. In this case, we can solve the equation x−6y=6x - 6y = 6 for xx:

x=6+6yx = 6 + 6y

Now, we can substitute this expression for xx into the equation x−6y=6x - 6y = 6:

(6+6y)−6y=6(6 + 6y) - 6y = 6

Simplifying the equation, we get:

6=66 = 6

This is a true statement, which means that the equation x−6y=6x - 6y = 6 is an identity. However, this does not mean that we have found a solution to the equation. Instead, it means that the equation is true for all values of xx and yy.

Finding a Particular Solution

Since the equation x−6y=6x - 6y = 6 is an identity, we can choose any values for xx and yy and substitute them into the equation. Let's choose x=12x = 12 and y=1y = 1. Substituting these values into the equation, we get:

12−6(1)=612 - 6(1) = 6

Simplifying the equation, we get:

12−6=612 - 6 = 6

This is a true statement, which means that the ordered pair (12,1)(12, 1) is a solution to the equation x−6y=6x - 6y = 6.

Conclusion

In this article, we solved the linear equation x−6y=6x - 6y = 6 using the method of substitution. We found that the equation is an identity, which means that it is true for all values of xx and yy. However, we also found a particular solution to the equation, which is the ordered pair (12,1)(12, 1). This solution satisfies the equation, and it can be used as a starting point for further calculations.

Final Answer

The final answer is:

(x,y)=(12,1)(x, y) = \boxed{(12, 1)}

Note: The final answer is an ordered pair, which means that it consists of two values, xx and yy. In this case, the ordered pair is (12,1)(12, 1), which means that x=12x = 12 and y=1y = 1.

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Introduction

In our previous article, we solved the linear equation x−6y=6x - 6y = 6 using the method of substitution. We found that the equation is an identity, which means that it is true for all values of xx and yy. However, we also found a particular solution to the equation, which is the ordered pair (12,1)(12, 1). In this article, we will answer some common questions related to solving linear equations.

Q&A

Q: What is a linear equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, a linear equation is an equation that can be written in the form ax+by=cax + by = c, where aa, bb, and cc are constants, and xx and yy are variables.

Q: How do I solve a linear equation?

There are several methods for solving linear equations, including substitution, elimination, and graphing. The method you choose will depend on the specific equation and the values of the variables.

Q: What is the difference between a solution and a particular solution?

A solution to a linear equation is a value or set of values that satisfies the equation. A particular solution is a specific solution that satisfies the equation, but may not be the only solution.

Q: How do I find a particular solution to a linear equation?

To find a particular solution to a linear equation, you can choose values for the variables and substitute them into the equation. If the equation is true for those values, then you have found a particular solution.

Q: What is the significance of the ordered pair (x, y) in a linear equation?

The ordered pair (x, y) represents the values of the variables that satisfy the equation. In other words, it represents the solution to the equation.

Q: Can I use a calculator to solve a linear equation?

Yes, you can use a calculator to solve a linear equation. Many calculators have built-in functions for solving linear equations, such as the "solve" function.

Q: How do I check my solution to a linear equation?

To check your solution to a linear equation, you can substitute the values of the variables into the equation and simplify. If the equation is true, then you have found a correct solution.

Conclusion

In this article, we answered some common questions related to solving linear equations. We discussed the definition of a linear equation, the methods for solving linear equations, and the significance of the ordered pair (x, y) in a linear equation. We also provided some tips for finding a particular solution to a linear equation and checking your solution.

Final Tips

  • Always read the problem carefully and understand what is being asked.
  • Choose a method for solving the equation that you are comfortable with.
  • Check your solution to make sure it is correct.
  • Use a calculator if you need to simplify the equation or find a particular solution.

Common Mistakes

  • Not reading the problem carefully and understanding what is being asked.
  • Choosing the wrong method for solving the equation.
  • Not checking the solution to make sure it is correct.
  • Not using a calculator when necessary.

Final Answer

The final answer is:

  • A linear equation is an equation in which the highest power of the variable(s) is 1.
  • The method you choose for solving a linear equation will depend on the specific equation and the values of the variables.
  • A particular solution is a specific solution that satisfies the equation, but may not be the only solution.
  • The ordered pair (x, y) represents the values of the variables that satisfy the equation.
  • You can use a calculator to solve a linear equation and check your solution.

Note: The final answer is a summary of the key points discussed in the article.