Download Math 111 (071) Fall 2008 Examples 1

Math 111 (071) Fall 2008
Examples 1 - 14
Example 1
Graph the linear equation 5x − 3y = 30. What are the x and y-intercepts of the line?
Example 2
Which of the following equations describe the same line as the equation 5x − 3y = 30?
(a)
6 + 53 x − y = 0
(b) 6 − x + 53 y = 0
(d) y = 53 x − 10
(e)
x = − 53 y + 10
(c)
y = − 53 x + 10
(f)
3
5y −
x=6
Example 3
L3
Each of the lines L1 , L2 , and L3 shown is the graph of one of the
equations (a), (b), or (c). Match each of the equations with its
corresponding line.
(a)
y = x+2
(b) 4x + 5y = 20
(c)
b
(0, 4)
2x + 5y = 10
(0, 2)
b
(−2, 0)
b
b
(5, 0)
L2
L1
Example 4
Suppose that a manufacturer finds that the cost y of producing x units of a certain commodity is given by
the linear equation
y = 700 + 30x
(a) Graph this linear equation.
(b) What is the y-intercept and what does it represent in this problem?
(c) Does the x-intercept of the line have any useful interpretation in this problem?
(d) What is the cost of producing 50 units of the commodity?
(e) How many units are produced if the cost is $3100?
Example 5
Solve each inequality and graph the solution.
(a)
2x + 5 ≤ − x + 14
(b) 3x − 8 > 2( x − 12)
Example 6
Graph the linear inequality.
(a)
3x − 5y ≤ 15
(b) 4x + 3y < 10
Example 7
Graph each system of linear inequalities.
2x + 3y
≥5
x + 3y
(a)
(b)
4x − 5y ≤ 21
x + 3y
≤ 10
≥6
(c)
y≤4
(d) x ≥ 5
Math 111 (071)
Fall 2008
Examples 1 - 14
Page 2
Example 8
Solve each system of linear equations and interpret your answer as the point of intersection of two lines.
3x − 4y =
1
x + 3y = 2
(b)
(a)
x + 2y = −3
2x + 5y = 8
4x + 3y = 8
2x − 3y =
8
(d)
(c)
y = 2
−6x + 9y = 12
Example 9
Graph the system of linear inequalities and find all of the vertices of the feasible set.




 x + 2y ≥ 8
 x + 2y ≤ 40


x+y ≥ 5
5x + 2y ≤ 80
(b)
(a)
x ≥2
x
≥
0






y ≥0
y
≥0
Example 10
Find an equation for and draw the line:
(a) that goes through the points (−4, −2) and (4, 4).
3
(b) that goes through the points (1, 3) and has slope m = − .
2
Example 11
The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800
to produce 300 chairs in one day.
(a) Assuming that the relationship between cost and the number of chairs produced is linear, find an
equation that expresses this relationship. Graph the equation on a set of axes that makes sense for this
problem.
(b) What is the slope of the line in part (a), and what does it represent?
(c) What is the y-intercept of this line, and what does it represent?
Example 12
The supply curve for a commodity relates the number of units of the commodity that will be produced, q
for quantity, and the price p, in dollars, that is charged per unit. The demand curve relates the number of
units of the commodity that can be sold, q, and the price p. For a certain commodity the supply curve is
a straight line with equation p = 0.0025q + 5, and the demand curve is also a straight line with equation
p = 8 − 0.0035q.
(a) The supply and demand equations above give the price p in terms of the quantity q. Give the slope
of each linear equation and explain what it means in terms of supply and demand. Determine any
intercepts that are relevant and explain what they mean in terms of supply and demand.
(b) Write the supply and demand equations giving q in terms of p. Give the slope of each linear equation
and explain what it means in terms of supply and demand. Determine any intercepts that are relevant
and explain what they mean in terms of supply and demand.
(c) Find the number of units of the commodity that will be sold and the price at which it will sell when
the market for this commodity reaches equilibrium.
Example 13
Find the slope and y-intercepts of each line and graph the line.
(a)
2x + 3y = 6
(b)
x = −2
(c)
2y = 5y − 9
Example 14
Find an equation in standard form (slope-intercept form) and in general form of the line parallel to the line
through the points (−3, 4) and (−3, −2) that contains the point (4, 2)