# Linear Equations and Graphing

**Special forms for linear equations :
Slope- Intercept form .**

Definition 4 An equation of the form

y = mx+b

is a linear equation in slope-intercept form.

Put 4x − 3y = 12 into slope- intercept form .

What are m and b?

What do they mean?

Price-demand equation: d = 1720 − .50p.

Put this into slope-intercept form.

What are m and b?

What is the meaning of m? of b?

In general, the graph of an equation y = mx + b defines a line

with slope m and y -intercept (0, b).

**Special forms for linear equations : Point Slope
form.**

Definition 5 An equation of the form

y − y_{1} = m(x − x_{1})

is a linear equation in point-slope form.

In general, the graph of an equation y − y_{1} = m(x − x_{1}) is a line

with slope m passing through the point (x_{1}, y_{1}).

Example: What is the point-slope equation for the line with a

slope of 3 passing through the point (−4, 6)?

In this case m = 3 and (x_{1}, y_{1}) = (−4, 6). So the point-slope

form for the equation of the line is

y − 6 = 3(x+4).

The graph of an equation y −y_{1} = m(x−x_{1}) is a line with slope

m passing through the point (x_{1}, y_{1}).

Example: Find an equation for the line passing through the

points (2, 1) and (5, 3).

To use the point-slope form for the line, we need to know one

point and the slope. In this problem the slope is not given, but

two points are known. To find the slope m, we use the fact that

So the point-slope form for the equation using (2, 1) as the point

is

If we use (5, 3) instead, then the point-slope form of the equation

is

These are two different equations for the same line.

**Application: Depreciation.**

Linear Depreciation. Office equipment was purchased for $20,000

and is assumed to have a scrap value of $2,000 after 10 years. If

its value is depreciated linearly (for tax purposes) from $20,000

to $2,000:

1. Find the linear equation that relates value (V) in dollars to

time (t) in years. (Hint: you know two points .)

1. Ans: V = −1800t + 20000. Write a verbal interpretation of

the slope of the line.

2. What would be the value of the equipment after 6 years?

3. Graph the equation V = −1800t+20000 for

**Application: Linear Interpolation**

The price of a cup of coffee in a coffee bar depends on the size of

the cup. The 8-ounce cup costs $2.10, but the larger 20-ounce

cup costs $3.30. Without any other information, how could you

estimate the cost of a 10-ounce cup, or a 16-ounce cup?

Sometimes business people use a method known as linear interpolation.

This means that they assume that the price p of a

cup of coffee and the size q of the cup obey a linear equation.

So the graph of the linear equation is a line and we know two

points on this line: (q, p) = (8, 2.10), (20, 3.30).

Units: The price p of the cup of coffee is given in dollar.

Size q of the cup of coffee is given in ounces.

two points on the line: (q, p) = (8, 2.10), (20, 3.30).

Slope: We know that the price of an 8-ounce cup is $2.10 and

the price of a 20-ounce cup is $3.30. So the change in q is

$3.30−$2.10 = $1.20 and the change in p is 20oz.−8oz. = 12oz..

So the slope is:

Units for the slope m: dollars per ounce

**Application: Linear Interpolation
**Slope is $.10 per ounce.

How much does the price increase if the size of the cup increases

by 4 ounces?

Point-slope form: p − 2.10 = .10(q − 8)

Slope-intercept form: p = .10q +1.30.

y-intercept is $1.30.

Interpretation: The price of a cup of coffee is $1.30 plus ten

cents an ounce.

Slope-intercept form: p = .10q +1.30.

Use linear interpolation to find the price of a 12-ounce cup of

coffee:

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