Find Value Of K For Point (1, -7) As Solution To Equation

#Introduction

In mathematics, particularly in algebra, we often encounter linear equations and the concept of solutions to these equations. A solution to a linear equation is a set of values for the variables that, when substituted into the equation, make the equation true. In this article, we will explore how to determine the value of a constant, denoted as k, in a linear equation, given a specific point that is a solution to the equation. The problem at hand involves finding the value of k for which the point (1, -7) is a solution of the equation ${ \frac{2}{3}x - \frac{7}{4}y = k }$. This is a fundamental problem in algebra that helps us understand the relationship between points and linear equations. By understanding how to solve this type of problem, we will also gain valuable insights into the nature of linear equations and their graphical representations.

Understanding Linear Equations and Solutions

Before we dive into the specific problem, let's establish a clear understanding of linear equations and their solutions. A linear equation in two variables, such as x and y, can be generally represented in the form ${ ax + by = c }$, where a, b, and c are constants. A solution to such an equation is an ordered pair (x, y) that satisfies the equation when the values of x and y are substituted into the equation. Graphically, a linear equation represents a straight line on the coordinate plane, and the solutions to the equation correspond to the points that lie on this line. Therefore, if a point (x, y) is a solution to the equation, it means that this point lies on the line represented by the equation. When we are given a specific point and a linear equation with an unknown constant, like k in our case, we can use the fact that the point is a solution to find the value of the constant. This is done by substituting the coordinates of the point into the equation and solving for the unknown constant. This process essentially involves plugging in the given x and y values into the equation and then performing algebraic manipulations to isolate and find the value of k. This method is a cornerstone of solving many problems in algebra and is crucial for understanding the behavior of linear equations and their solutions.

Problem Statement and Solution Strategy

Our main task is to determine the value of k in the equation ${ \frac{2}{3}x - \frac{7}{4}y = k }$, given that the point (1, -7) is a solution. To solve this problem, we will employ a straightforward substitution method. The strategy is as follows:

1. Substitute the Coordinates: Replace x and y in the equation with the given coordinates of the point (1, -7). This means substituting x = 1 and y = -7 into the equation.
2. Simplify the Equation: After substitution, we will simplify the equation by performing the necessary arithmetic operations, such as multiplication and addition/subtraction. This will help us isolate k on one side of the equation.
3. Solve for k: The final step involves solving the simplified equation for k. This will give us the value of k for which the point (1, -7) is a solution of the given equation. By following these steps, we can systematically find the value of k. This method highlights the fundamental connection between algebraic equations and their solutions, providing a clear and logical approach to solving the problem.

Step-by-Step Solution

Let's now proceed with the step-by-step solution to find the value of k:

1. Substitute the Coordinates

We are given the equation: ${ \frac{2}{3}x - \frac{7}{4}y = k }$

And the point (1, -7). We substitute x = 1 and y = -7 into the equation: ${ \frac{2}{3}(1) - \frac{7}{4}(-7) = k }$

This substitution replaces the variables x and y with their respective values from the given point, allowing us to transform the equation into a simpler form that involves only the unknown constant k.

2. Simplify the Equation

Next, we simplify the equation by performing the arithmetic operations: ${ \frac{2}{3} + \frac{49}{4} = k }$

Here, we multiplied the fractions. Now, we need to find a common denominator to add the fractions. The least common multiple (LCM) of 3 and 4 is 12. So, we rewrite the fractions with the common denominator: ${ \frac{2}{3} \times \frac{4}{4} + \frac{49}{4} \times \frac{3}{3} = k }$ ${ \frac{8}{12} + \frac{147}{12} = k }$

Now that the fractions have the same denominator, we can add them: ${ \frac{8 + 147}{12} = k }$ ${ \frac{155}{12} = k }$

3. Solve for k

From the simplified equation, we have: ${ k = \frac{155}{12} }$

Thus, the value of k for which the point (1, -7) is a solution of the equation ${ \frac{2}{3}x - \frac{7}{4}y = k }$ is ${ \frac{155}{12} }$. This final result gives us the specific value of k that makes the given point a valid solution to the linear equation, completing the problem-solving process.

Verification

To ensure our solution is correct, we can verify it by substituting the values of x, y, and k back into the original equation: ${ \frac{2}{3}x - \frac{7}{4}y = k }$

Substitute x = 1, y = -7, and k = \frac{155}{12}: ${ \frac{2}{3}(1) - \frac{7}{4}(-7) = \frac{155}{12} }$

Simplify the left side of the equation: ${ \frac{2}{3} + \frac{49}{4} = \frac{155}{12} }$

Find a common denominator (12): ${ \frac{8}{12} + \frac{147}{12} = \frac{155}{12} }$

Add the fractions: ${ \frac{155}{12} = \frac{155}{12} }$

The left side of the equation equals the right side, which confirms that our solution for k is correct. This verification step is crucial in ensuring the accuracy of our result and provides a solid confirmation that the calculated value of k indeed makes the given point a solution to the original equation.

Conclusion

In this article, we successfully found the value of k for which the point (1, -7) is a solution of the equation ${ \frac{2}{3}x - \frac{7}{4}y = k }$. By substituting the coordinates of the point into the equation, simplifying, and solving for k, we determined that k = \frac{155}{12}. We also verified our solution to ensure its correctness. This exercise demonstrates a fundamental concept in algebra: how to find unknown constants in linear equations given a solution point. This method is widely applicable in various mathematical contexts and is essential for understanding the relationships between equations and their solutions. Mastering these skills is crucial for further studies in mathematics and related fields.

This problem highlights the importance of understanding linear equations and their solutions. The ability to find unknown constants given a solution is a valuable skill in algebra and has practical applications in various fields, including engineering, physics, and computer science. By mastering these fundamental concepts, students can build a strong foundation for more advanced mathematical topics.