Determining the Value of x for a Specific Line Slope

In the field of geometry, understanding the relationship between coordinates and the slope of a line is fundamental. This article will explore the value of x when a line goes through certain points with a specified slope. Let's dive into the intricacies of this problem and find the answer.

Problem Analysis

The problem at hand is to determine the value of x such that the line passing through the points (3, x) and (-2, 3) has a slope of 1.

Direct Method: Using the Slope Formula

The slope (m) of a line between two points ((x_1, y_1)) and ((x_2, y_2)) is given by:

Slope Formula: (m frac{y_2 - y_1}{x_2 - x_1})

Given the points (3, x) and (-2, 3), we can substitute these into the slope formula:

Slope Calculation:

[m frac{3 - x}{-2 - 3} frac{3 - x}{-5}]

For the slope to be 1, we set (m 1):

[1 frac{3 - x}{-5}]

Solving for (x), we get:

[1 frac{3 - x}{-5}]

[-5 3 - x]

[x 8]

This indicates that when the line has a slope of 1, the value of (x) is 8.

Alternative Method: Using the Slope-Intercept Form

The slope-intercept form of a line is given by:

Slope-Intercept Form: (y mx c)

Here, (m) is the slope and (c) is the y-intercept. Given that the slope (m 1), the equation becomes:

[y x c]

Since the point (-2, 3) lies on the line, we can substitute these values:

[3 -2 c]

Solving for (c), we get:

[c 5]

Thus, the equation of the line with slope 1 through (-2, 3) is:

[y x 5]

This confirms that the value of (x) must be 3, as the point (3, 8) lies on this line, giving a slope of 1.

Conclusion: Importance of Coordinates

It is crucial to understand that both the x and y coordinates play a significant role in determining the line equation and its slope. In this problem, the value of (x) is calculated by ensuring the slope of the line is 1, and the coordinates provided need to fit this condition.

In summary, the value of (x) for the line passing through (3, x) and (-2, 3) with a slope of 1 is 8. This problem highlights the importance of using the correct slope formula and slope-intercept form to solve similar geometry problems.