For The Equation \( \frac{1}{4}x = Cx + 2 \), Where \( C \) Is A Constant, Find The Value Of \( C \) That Results In The Equation Having No Solution.

In the realm of mathematics, solving equations is a fundamental skill. We often encounter linear equations, which are algebraic expressions where the highest power of the variable is one. These equations can have one solution, infinitely many solutions, or no solution at all. In this comprehensive article, we delve into the specific scenario of a linear equation with one variable, exploring how to determine the value of a constant that results in the equation having no solution. Our focus is on the equation ${ \frac{1}{4}x = cx + 2 }$, where ${ c }$ is a constant. We aim to find the precise value of ${ c }$ that makes this equation unsolvable.

Before diving into the problem at hand, it's crucial to grasp the concept of solutions in linear equations. A solution to an equation is a value of the variable that makes the equation true. Graphically, a linear equation represents a straight line. The solution to a system of linear equations corresponds to the point(s) where the lines intersect. When two lines are parallel, they never intersect, indicating that the system has no solution. In the context of a single linear equation, if the variable terms cancel out and leave a contradiction (e.g., 0 = 2), then the equation has no solution.

To make it more understandable, let’s consider a simple linear equation: ax + b = 0. This equation has a unique solution if a is not equal to zero. The solution is given by x = -b/a. However, if a = 0, the equation becomes 0x + b = 0, which simplifies to b = 0. If b is also zero, then the equation is true for any value of x, implying infinitely many solutions. But if b is not zero, we have a contradiction (e.g., 0 = 2), indicating that there is no solution. This fundamental understanding of how coefficients affect solutions is key to solving our given problem.

The given equation ${ \frac{1}{4}x = cx + 2 }$ is a linear equation in the variable ${ x }$. The constant ${ c }$ plays a crucial role in determining the nature of the solutions. Our goal is to find the specific value of ${ c }$ that makes the equation have no solution. This occurs when the coefficients of ${ x }$ on both sides of the equation are equal, while the constant terms are different. This situation creates a contradiction, preventing any value of ${ x }$ from satisfying the equation. By carefully manipulating the equation and analyzing the conditions for no solution, we can pinpoint the exact value of ${ c }$ that fulfills this criterion. The following sections will guide you through the step-by-step process of solving the equation and identifying the condition for no solution.

Solving for No Solution

To determine the value of ${ c }$ for which the equation ${ \frac{1}{4}x = cx + 2 }$ has no solution, we need to manipulate the equation to isolate ${ x }$ terms on one side and constant terms on the other. The first step is to subtract ${ cx }$ from both sides of the equation. This gives us:

${ \frac{1}{4}x - cx = 2 }$

Next, we factor out ${ x }$ from the left side of the equation:

${ x(\frac{1}{4} - c) = 2 }$

Now, we analyze the equation to find the condition for no solution. An equation has no solution when the coefficient of ${ x }$ is zero, and the constant term is non-zero. In this case, the coefficient of ${ x }$ is ${ \frac{1}{4} - c }$, and the constant term is 2. For the equation to have no solution, we need:

${ \frac{1}{4} - c = 0 }$

Solving this equation for ${ c }$, we add ${ c }$ to both sides:

${ \frac{1}{4} = c }$

Thus, we find that ${ c = \frac{1}{4} }$. Now, let's check what happens when we substitute this value back into the original equation:

${ \frac{1}{4}x = \frac{1}{4}x + 2 }$

Subtracting ${ \frac{1}{4}x }$ from both sides, we get:

${ 0 = 2 }$

This is a contradiction, which confirms that the equation has no solution when ${ c = \frac{1}{4} }$. This step-by-step approach ensures that we identify the exact value of ${ c }$ that leads to an unsolvable equation. By understanding the conditions under which linear equations have no solution, we can confidently tackle similar problems and apply these concepts to more complex scenarios. The ability to manipulate equations and identify contradictions is a crucial skill in mathematics, especially when dealing with linear algebra and systems of equations.

Graphical Interpretation

A graphical interpretation can provide a deeper understanding of why the equation has no solution when ${ c = \frac{1}{4} }$. Let’s rewrite the original equation ${ \frac{1}{4}x = cx + 2 }$ as two separate linear equations:

${ y = \frac{1}{4}x }$

${ y = cx + 2 }$

These equations represent two straight lines. The solutions to the original equation are the points where these two lines intersect. When ${ c = \frac{1}{4} }$, the second equation becomes:

${ y = \frac{1}{4}x + 2 }$

Now we have two lines:

${ y = \frac{1}{4}x }$

${ y = \frac{1}{4}x + 2 }$

Notice that these two lines have the same slope (${ \frac{1}{4} }$) but different y-intercepts (0 and 2, respectively). This means the lines are parallel and will never intersect. Parallel lines, by definition, have the same slope but different y-intercepts. Since the lines do not intersect, there is no point ${ (x, y) }$ that satisfies both equations simultaneously. Therefore, the original equation has no solution when ${ c = \frac{1}{4} }$. This graphical perspective reinforces the algebraic solution, providing a visual representation of why the equation is unsolvable. Understanding the connection between algebraic manipulations and graphical interpretations is crucial for a comprehensive understanding of linear equations.

Visualizing Parallel Lines

To visualize this, imagine plotting these two lines on a coordinate plane. The first line, ${ y = \frac{1}{4}x }$, passes through the origin (0,0) and has a gentle upward slope. The second line, ${ y = \frac{1}{4}x + 2 }$, is parallel to the first line but is shifted upwards by 2 units on the y-axis. No matter how far you extend these lines, they will never meet, confirming that there is no solution to the equation. This graphical representation is a powerful tool for understanding concepts in algebra and calculus. It allows us to see abstract equations as concrete geometric objects, making the concepts more intuitive and easier to grasp. The relationship between algebra and geometry is a fundamental aspect of mathematics, and understanding this relationship can greatly enhance problem-solving skills.

Conclusion

In conclusion, we have successfully determined the value of ${ c }$ for which the equation ${ \frac{1}{4}x = cx + 2 }$ has no solution. By manipulating the equation algebraically, we found that ${ c = \frac{1}{4} }$ results in a contradiction, indicating that no value of ${ x }$ can satisfy the equation. Furthermore, we provided a graphical interpretation, demonstrating that when ${ c = \frac{1}{4} }$, the equation represents two parallel lines, which never intersect, thus confirming the absence of a solution. This exercise highlights the importance of understanding the conditions under which linear equations have no solution, as well as the connection between algebraic and graphical methods in solving mathematical problems. Mastering these concepts is crucial for further studies in mathematics and related fields.

• Original Question: Given the equation ${ \frac{1}{4}x = cx + 2 }$, where ${ c }$ is a constant, what is the value of ${ c }$ such that the equation has no solution?
• Repaired Question: For the equation ${ \frac{1}{4}x = cx + 2 }$, where ${ c }$ is a constant, find the value of ${ c }$ that results in the equation having no solution.

Find the Value of c for No Solution in the Equation (1/4)x = cx + 2