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Linear Equations. Slope, Intercepts, Applications
•Linear Equations in two variables
•Slope of a line
•Intercepts
•y = mx+b
•Applications
x and yintercepts of an equation.
Definition In the graph of any equation of two variables, the
points where the graph of an equation crosses the xaxis are
called the xintercepts and the points where the graph crosses
the yaxis are called the yintercepts
What do the coordinates of an xintercept look like?
To find them set = 0 and solve for .
What do the coordinates of a yintercept look like?
To find them set = 0 and solve for .
The intercepts of a linear equation are easier to locate than the
intercepts of most other equations.
x and yintercepts of a linear equation.
Find the x and y intercepts for 4x  3y = 12.
Find the x and yintercepts for 7x  .2y = 12.
Could a linear equation have more than one xintecept?
Intercepts for the Pricedemand equation.
d = 1720  .50p
What are the p and dintercepts? What do they mean here?
The slope of a linear equation. the pricedemand
equation
d = 1720  .50p
If price increases by $1 how much does demand decrease?
Does this depend on the starting price?
If price increases by $1000 how much does demand decrease?
Does this depend on the starting price?
Is the decrease in demand always .50 times the increase in price?
SlopeIntercept form (y = mx+b).
Definition An equation of the form
y = mx+b
is a linear equation in slopeintercept form.
Put 4x  3y = 12 into slopeintercept form.
What are m and b?
Give a verbal description of m and b.
Pricedemand equation. d = 1720  .50p.
Put this into slopeintercept form.
What are m and b?
What is the meaning of m? of b?
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. What is the slope of the line? Write a verbal interpretation
of the slope of the line.
2. Find the linear equation that relates value (V) in dollars to
time (t) in years. (Hint. you know two points.)
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. What would be the value of the equipment after 6 years?
2. Graph the equation
Application. Linear Interpolation
The price of a cup of coffee in a coffee bar depends on the size of
the cup. The 8ounce cup costs $2.10, but the larger 20ounce
cup costs $3.30. Without any other information, how could you
estimate the cost of a 10ounce cup, or a 16ounce 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).
Application. Linear Interpolation
two points on the line. (q, p) = (8, 2.10), (20, 3.30).
Slope.
Pointslope form. p  2.10 = .10(q  8)
Slopeintercept form. p = .10q +1.30.
Application. Linear Interpolation
Slopeintercept form. p = .10q +1.30.
Use linear interpolation to find the price of a 12ounce cup of
coffee.
Give a verbal interpretation for the slope.
If the size of the cup increases by 4 ounces, by how much does
the cost increase?
Give a verbal interpretation for the yintercept.
Application. So Relax
West field shopping mall has an accupressure massage station
called "So Relax." Here is the pricing structure.
10 minutes $13.00
15 minutes $19.00
20 minutes $26.00
•Plot these prices
•Is the price a linear function
of the time in minutes?
•What is the best deal?