The Point-Slope Form Formula -

Formula

y-y_1 = m(x-x_1)

Used to calculate:

A linear equation from the slope of a line and the coordinates of a known point on the line.

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What Is the Point-Slope Form Formula?

The point-slope form is a way to represent a straight line using its slope and the coordinates of one known point on the line. It is especially useful when the slope and a point are known but the y-intercept is not.

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Formula and Variables

y-y_1 = m(x-x_1)

Symbol	Meaning	Unit
$x$	x-coordinate of any point on the line	Depends on the problem
$y$	y-coordinate corresponding to x	Depends on the problem
$x_1$	x-coordinate of the known point	Same unit as x
$y_1$	y-coordinate of the known point	Same unit as y
$m$	Slope of the line	Units of y per unit of x

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How to Use the Formula

Identify a known point on the line and its coordinates ( $x_1$ , $Y_1$ ), along with the slope m. Substitute these values into the point-slope formula. The resulting equation represents the line and can be rearranged into another linear form, such as slope-intercept form, if needed.

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When Is This Formula Used?

Point-slope form is used to write or analyze the equation of a nonvertical straight line when its slope and at least one point on the line are known.

Common uses:

– Writing the equation of a line from a point and its slope
– Constructing a line from two known points after calculating the slope
– Converting a linear equation into other forms

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Important Notes

– The point ( $x_1$ , $y_1$ ) must lie on the line represented by the equation.
– The standard point-slope form assumes a defined finite slope, so it does not represent vertical lines, which instead have equations of the form $x=x_1$ .
– The quantities x and $x_1$ must have compatible units, as must y and $y_1$ ; slope has units of y per unit of x.
– A common point of confusion is mishandling negative coordinates. Because the formula already contains subtraction, substituting a negative coordinate produces subtraction of a negative number.

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Alternate Forms

Alternate form:

y-y_1 = mx-mx_1

Alternate form:

y = mx + (y_1-mx_1)

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