Complete The Table By Finding The Values Of \[$x\$\] And \[$y\$\] That Satisfy The Equation.Equation: \[$x - 4y = 4\$\]$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & \\ \hline & 0 \\ \hline &
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Introduction
In this article, we will explore the concept of solving a linear equation in two variables. A linear equation in two variables is an equation that can be written in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. We will use the given equation x - 4y = 4 to find the values of x and y that satisfy the equation.
The Equation
The given equation is x - 4y = 4. This is a linear equation in two variables, where the coefficients of x and y are 1 and -4, respectively, and the constant term is 4.
Finding the Values of x and y
To find the values of x and y that satisfy the equation, we can use the method of substitution or elimination. In this case, we will use the method of substitution.
Substitution Method
We can start by isolating one of the variables in the equation. Let's isolate y by adding 4y to both sides of the equation:
x - 4y + 4y = 4 + 4y
This simplifies to:
x = 4 + 4y
Now, we can substitute this expression for x into the original equation:
x - 4y = 4
Substituting x = 4 + 4y into the equation, we get:
(4 + 4y) - 4y = 4
Simplifying this equation, we get:
4 = 4
This is a true statement, which means that the equation is an identity. This means that the equation is true for all values of x and y.
Discussion
Since the equation is an identity, we can conclude that the values of x and y are not unique. In other words, there are infinitely many solutions to the equation.
Solving for x and y
To find the values of x and y that satisfy the equation, we can use the fact that the equation is an identity. This means that we can choose any values for x and y that satisfy the equation.
Example 1
Let's choose x = 0 and y = 0. Substituting these values into the equation, we get:
0 - 4(0) = 4
This simplifies to:
0 = 4
This is a false statement, which means that the values x = 0 and y = 0 do not satisfy the equation.
Example 2
Let's choose x = 4 and y = 0. Substituting these values into the equation, we get:
4 - 4(0) = 4
This simplifies to:
4 = 4
This is a true statement, which means that the values x = 4 and y = 0 satisfy the equation.
Example 3
Let's choose x = 0 and y = 1. Substituting these values into the equation, we get:
0 - 4(1) = 4
This simplifies to:
-4 = 4
This is a false statement, which means that the values x = 0 and y = 1 do not satisfy the equation.
Example 4
Let's choose x = 4 and y = 1. Substituting these values into the equation, we get:
4 - 4(1) = 4
This simplifies to:
0 = 4
This is a false statement, which means that the values x = 4 and y = 1 do not satisfy the equation.
Conclusion
In conclusion, we have shown that the equation x - 4y = 4 is an identity, which means that there are infinitely many solutions to the equation. We have also shown that the values of x and y are not unique, and that there are many possible combinations of x and y that satisfy the equation.
Final Answer
The final answer is that the equation x - 4y = 4 has infinitely many solutions, and that the values of x and y are not unique.
Discussion Category
The discussion category for this article is mathematics.
Table of Values
| x | y |
|---|---|
| 0 | |
| 0 | |
| 4 | 0 |
| 0 | 1 |
| 4 | 1 |
Note: The table of values is incomplete, as we have not found the values of x and y that satisfy the equation. However, we have shown that the equation is an identity, which means that there are infinitely many solutions to the equation.
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Introduction
In our previous article, we explored the concept of solving a linear equation in two variables. We used the equation x - 4y = 4 to find the values of x and y that satisfy the equation. In this article, we will answer some frequently asked questions about solving linear equations in two variables.
Q&A
Q: What is a linear equation in two variables?
A: A linear equation in two variables is an equation that can be written in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: How do I solve a linear equation in two variables?
A: To solve a linear equation in two variables, you can use the method of substitution or elimination. In the case of the equation x - 4y = 4, we used the method of substitution to find the values of x and y that satisfy the equation.
Q: What is the difference between the method of substitution and the method of elimination?
A: The method of substitution involves isolating one of the variables in the equation and then substituting that expression into the original equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Q: Can I use the method of substitution to solve any linear equation in two variables?
A: Yes, you can use the method of substitution to solve any linear equation in two variables. However, you need to make sure that you can isolate one of the variables in the equation.
Q: What if I get a false statement when I substitute the values into the equation?
A: If you get a false statement when you substitute the values into the equation, it means that the values you chose do not satisfy the equation. You need to try different values until you find a combination that satisfies the equation.
Q: Can I use the method of elimination to solve any linear equation in two variables?
A: Yes, you can use the method of elimination to solve any linear equation in two variables. However, you need to make sure that the coefficients of the variables are the same in both equations.
Q: What if I get a true statement when I add or subtract the equations?
A: If you get a true statement when you add or subtract the equations, it means that the equations are dependent, and there are infinitely many solutions to the system.
Q: Can I use a graph to solve a linear equation in two variables?
A: Yes, you can use a graph to solve a linear equation in two variables. However, you need to make sure that the graph is accurate and that you can find the point of intersection between the two lines.
Q: What if I get a point of intersection that is not a solution to the equation?
A: If you get a point of intersection that is not a solution to the equation, it means that the equations are not dependent, and there is only one solution to the system.
Conclusion
In conclusion, solving a linear equation in two variables can be a challenging task, but with the right techniques and strategies, you can find the values of x and y that satisfy the equation. Remember to use the method of substitution or elimination, and to check your work to make sure that you have found the correct solution.
Final Answer
The final answer is that solving a linear equation in two variables requires patience, persistence, and practice. With these skills, you can master the art of solving linear equations in two variables and become a proficient mathematician.
Discussion Category
The discussion category for this article is mathematics.
Table of Values
| x | y |
|---|---|
| 0 | |
| 0 | |
| 4 | 0 |
| 0 | 1 |
| 4 | 1 |
Note: The table of values is incomplete, as we have not found the values of x and y that satisfy the equation. However, we have shown that the equation is an identity, which means that there are infinitely many solutions to the equation.