Understanding the Equation of a Line Given a Point and Angle

Are you interested in determining the equation of a line that passes through a specific point and makes a certain angle with the x-axis? If so, you're in the right place! This guide will take you through step-by-step to solve the problem of finding the equation of a line that passes through the point (7,3) and makes an angle of 45 degrees with the positive x-axis. Let's dive into it!

Step-by-Step Solution

Determining the Slope

To find the equation of the line, we first need to determine its slope. The slope of a line that makes an angle (theta) with the x-axis is given by the tangent of that angle:

(m tan{theta})

In this case, (theta 45^circ). Therefore, we can calculate the slope as follows:

(m tan{45^circ} 1)

Using the Point-Slope Form

The slope-intercept form of a line is (y mx b). However, we can also use the point-slope form, which is particularly useful when we know a point on the line. The point-slope form is given by:

(y - y_0 m(x - x_0))

In this problem, we know that the line passes through the point (7,3). Substituting the slope (m 1) and the point (7,3) into the point-slope form, we get:

(y - 3 1(x - 7))

Simplifying this equation, we have:

(y - 3 x - 7)

(y x - 4)

Conclusion

Thus, the equation of the line that passes through the point (7,3) and makes an angle of 45 degrees with the positive x-axis is:

(y x - 4)

Understanding the relationship between the angle of inclination and the slope of a line is a fundamental concept in linear algebra and geometry. This knowledge can be applied to various real-world problems, such as determining the slope of a road or the angle of a ramp.