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# 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.

## 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.

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.

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:

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

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

### 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.]

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

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

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

(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:

6x+3y+2=0
2x-4y=3

Perpendicular lines

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

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