Finding the Equation of a Line with Given Point and Slope

When you need to find the equation of a line that passes through a specific point and has a given slope, you can use the point-slope form or the slope-intercept form. Both methods lead to the same result, but they offer different perspectives on how to approach the problem.

Point-Slope Form

The point-slope form of the equation of a line is a direct way to derive the line's equation using just one point and the slope. Given a point ((x_1, y_1)) and a slope (m), the point-slope form is:

Step-by-Step Explanation

Identify the point and the slope. In this case, the point is ((1, -6)) and the slope (m 7). Substitute (x_1 1), (y_1 -6), and (m 7) into the point-slope form formula: (y - y_1 m(x - x_1)) (y - (-6)) 7(x - 1) Simplify the equation: (y 6 7x - 7) Solve for (y) by moving constants to one side: (y 7x - 7 - 6) (y 7x - 13)

The equation of the line is (y 7x - 13).

Slope-Intercept Form

The slope-intercept form of a line is another common representation, given as (y mx c), where (m) is the slope and (c) is the y-intercept. Let's use the same given point and slope to derive the equation:

Step-by-Step Explanation

Identify the slope (m 7). Use the point ((1, -6)) to find the y-intercept (c). Substitute (x_1 1) and (y_1 -6) into the slope-intercept form: (-6 7(1) c) Solve for (c): (-6 7 c) (c -6 - 7) (c -13)

The equation of the line in slope-intercept form is (y 7x - 13).

Verification

To verify, let's substitute (x 1) and check if (y -6):

(y 7(1) - 13 7 - 13 -6)

This confirms that the line passes through the point ((1, -6)).

Conclusion

Both methods, the point-slope form and the slope-intercept form, are effective in finding the equation of a line given a point and a slope. Understanding these methods not only helps in solving geometric problems but also in setting up linear models.

Keywords: line equation, point-slope form, slope-intercept form