Which equation represents the point-slope form of a line with slope m passing through (x1, y1)?

Which equation represents the point-slope form of a line with slope m passing through (x1, y1)?

• A. Y - y1 = m(x - x1)
• B. Y = mx + b
• C. Ax + By = C
• D. Y = y1 + mX

Answer: (A)

The main idea here is that the point-slope form writes a line using its slope and a specific point it passes through. If the slope is m and the line goes through (x1, y1), the equation is y − y1 = m(x − x1). This form makes it immediate to plug in the known point and slope, and it shows exactly how moving along x from x1 changes y by m times that distance. This form is the best fit because it directly encodes both pieces of information given in the problem: the slope and the point the line passes through. A slope-intercept form, y = mx + b, involves the y-intercept b, which isn’t given here and doesn’t guarantee the line passes through the designated point. A general form like Ax + By = C is not expressed in terms of a slope and a specific point. The expression y = y1 + mX is not correct because it doesn’t subtract the x-coordinate x1, so it doesn’t ensure the line passes through (x1, y1) in general. As a quick check, you can derive the point-slope form from the slope-intercept form by using the point (x1, y1) on the line: starting with y = mx + b and substituting y1 = m x1 + b gives b = y1 − m x1, and substituting back yields y = m x + (y1 − m x1), which rearranges to y − y1 = m(x − x1).

The main idea here is that the point-slope form writes a line using its slope and a specific point it passes through. If the slope is m and the line goes through (x1, y1), the equation is y − y1 = m(x − x1). This form makes it immediate to plug in the known point and slope, and it shows exactly how moving along x from x1 changes y by m times that distance.

This form is the best fit because it directly encodes both pieces of information given in the problem: the slope and the point the line passes through. A slope-intercept form, y = mx + b, involves the y-intercept b, which isn’t given here and doesn’t guarantee the line passes through the designated point. A general form like Ax + By = C is not expressed in terms of a slope and a specific point. The expression y = y1 + mX is not correct because it doesn’t subtract the x-coordinate x1, so it doesn’t ensure the line passes through (x1, y1) in general.

As a quick check, you can derive the point-slope form from the slope-intercept form by using the point (x1, y1) on the line: starting with y = mx + b and substituting y1 = m x1 + b gives b = y1 − m x1, and substituting back yields y = m x + (y1 − m x1), which rearranges to y − y1 = m(x − x1).