2. The Straight Line

Slope-Intercept Form of a Straight Line

The slope-intercept form (otherwise known as "gradient, y-intercept" form) of a line is given by:

y = mx + b

This tells us the slope of the line is m and the y-intercept of the line is b.

Example 1

The line y = 2x + 4 has

  • slope `m = 2` and
  • y-intercept `b = 4`.

We do not need to set up a table of values to sketch this line. Starting at the y-intercept (`y = 4`), we sketch our line by going up `2` units for each `1` unit we go to the right (since the slope is `2` in this example).

To find the x-intercept, we let `y = 0`.

2x + 4 = 0

`x = -2`

We notice that this is a function. That is, each value of x that we have gives one corresponding value of y.

See more on Functions and Graphs.

Continues below

Point-Slope Form of a Straight Line

We need other forms of the straight line as well. A useful form is the point-slope form (or point - gradient form). We use this form when we need to find the equation of a line passing through a point (x1, y1) with slope m:

y − y1 = m(xx1)

Example 2

Find the equation of the line that passes through `(-2, 1)` with slope of `-3`.

Answer

We use:

`y-y_1=m(x-x_1)`

Here,

`x_1= -2`

`y_1= 1`

`m = -3`

So the required equation is:

`y-1=-3(x-(-2)=-3x-6`

`y=-3x-5`

We have left it in slope-intercept form. We can see the slope is -3 and the y-intercept is -5.

General Form of a Straight Line

Another form of the straight line which we come across is general form:

Ax + By + C = 0

It can be useful for drawing lines by finding the y-intercept (put `x = 0`) and the x-intercept (put `y = 0`).

We also use General Form when finding Perpendicular Distance from a Point to a Line.

Example 3

Draw the line 2x + 3y + 12 = 0.

Answer

If `x = 0`, we have: `3y + 12 = 0`, so `y = -4`.

If `y = 0`, we have: `2x + 12 = 0`, so `x = -6`.

So the line is:

12-1-2-3-4-5-612-1-2-3-4-5xy

The line 2x + 3y + 12 = 0.

Note that the y-intercept is `-4` and the x-intercept is `-6`.

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Exercises

1. What is the equation of the line perpendicular to the line joining (4, 2) and (3, -5) and passing through (4, 2)?

[Need a reminder? See the section on Slopes of Perpendicular Lines.]

Answer

The line joining `(4, 2)` and `(3, -5)` has slope `m=(-7)/(-1)=7` and is shown as a green dotted line.

123456-1-2-3-4-5-61234-1-2-3-4-5-6xy(4, 2)(3, −5)

Perpendicular lines.

We need to find the equation of the magenta (pink) line.

The line perpendicular to the green dotted line has slope `-1/7.`

The line through `(4, 2)` with slope `-1/7` has equation:

`y-2=-1/7(x-4)`

`=-x/7+4/7`

`y=-x/7+2 4/7`

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2. If `4x − ky = 6` and `6x + 3y + 2 = 0` are perpendicular, what is the value of `k`?

Answer

(2) The slope of 4xky = 6 can be calculated by re-expressing it in slope-intercept form:

`y=4/kx-6/k`

So we see the slope is `4/k`.

The slope of `6x + 3y + 2 = 0` can also be calculated by re-expressing it in slope-intercept form:

`y=(-6)/3x-2/3=-2x-2/3`

So we see the slope is `-2`.

For the lines to be perpendicular, we need

`4/kxx-2=-1`

This gives `k = 8`.

The resulting line is `4x-8y=6`, which we can simplify to `2x-4y=3.` Here's the graph of the situation:

1 2 -1 -2 -3 -4 -5 -6 1 2 3 4 -1 -2 -3 -4 -5 x y
`6x+3y+2=0`
`2x-4y=3`

Perpendicular lines

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Conic section: Straight line

Each of the lines and curves in this chapter are conic sections, which means the curves are formed when we slice a cone at a certain angle.

How can we obtain a straight line from slicing a cone?

double cone

We start with a double cone (2 right circular cones placed apex to apex):

conic section - straight line

If we slice the double cone by a plane just touching one edge of the double cone, the intersection is a straight line, as shown.