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Writing Linear Equations in Slope-Intercept Form

Slope-Intercept Form

The slope-intercept form is a special case of the point-slope form. The given point of the line (x₀ , y₀ ) lies on the y-axis, so x₀ = 0. This means that the equation is of the form y - y₀ = mx, or y = mx + y₀ . The equation is therefore given explicitly when both the y-intercept and the slope are known, and is simpler than the more general point-slope form. This is perhaps the most common and most important normal form for the equation of a line.

Example 1

Write the slope-intercept form of the equation for a line that goes through (0, 4) and has a slope of 5.

Solution

y - y₀	= m ( x - x₀ )	Point-slope form
y - 4	= 5x	Replace x₀ with 0, y₀ with 4, and m with 5.

Now add 4 to each side of this equation in order to express y in terms of x . The result is the following equation.

y = 5 x + 4

Point out that 5 is the slope of the line and that 4 is the y-intercept.

Example 2

Write the slope-intercept form of the equation for a line that goes through (0, -3) and has a slope of -1.

Solution

y - y₀	= m ( x - x₀ )	Point-slope form
y - ( -3)	= ( -1)x	Replace x₀ with 0, y₀ with -3 , and m with -1.
y + 3	= -x

Now subtract 3 from each side of this equation in order to express y in terms of x . The result is the following equation.

y = -x - 3

Point out that -1 is the slope of the line and that -3 is the y -intercept.

Notice that these equations are really much simpler than the general point-slope form, since there are fewer terms. Also note that in each case, the coefficient of x is the slope and the constant term is the y-intercept.

Key Idea An equation for a line is in slope-intercept form if it is of the form y = mx + b, where m is the slope of the line and b is the y-intercept.

Any line that is not vertical has an equation that is in slope-intercept form. A vertical line could not possibly have an equation in slopeintercept form, for the same reason that it cannot have an equation in point-slope form - the slope is undefined.