A Bunch of Questions on Linear Functions (and direct variation)
1.
|
Which of the ordered
pairs
(6, 1), (8, 0), (4, –2), (–4, 6)
are solutions for the
equation x + 2y = 8?
|
3.
|
Complete the ordered
pairs for the equation 2x + y = 10.
(5, ), ( , 10), (
, –2), (0, )
|
When y=10: 2x+10=10\implies x=0, so (0,10).
When y=-2: 2x-2=10\implies x=6, so (6,-2).
When x=0: 2\times 0+y=10\implies y=10, so (0,10).
4.
|
Find four solutions for
the equation 3x + 5y = 15.
|
Solution: Just use 4 values of x and find the corresponding y values.
When x=0, 3(0)+5y=15\implies y=3 so (0,3).
When x=1, 3(1)+5y=15\implies y=12/5 so (1,12/5).
When x=2, 3(2)+5y=15\implies y=9/5 so (2,9/5).
When x=3, 3(3)+5y=15\implies y=6/5 so (3,6/5).
5.
|
Graph 2x – y
= 4.
|
6.
|
Graph using the
intercept method: $x + 3y = 6$.
|
When y=0, x=6
When x=0, 3y=6\implies y=2
So the x and y intercepts are x=6 and y=2.
7.
|
Graph by first solving
for y.
$4x – 3y = 6$
|
$4x – 3y = 6$
$– 3y = 6-4x$
$3y = -6+4x$
$3y =4x-6$
8.
|
Graph using the
intercept method: 2x + y = 4.
|
When y=0, 2x=4\implies x=2
When x=0, y=4
So the x and y intercepts are x=2 and y=4.
9.
|
Find the slope of the
line passing through the points (9, 12) and (8, 4).
|
Solution
Slope =\displaystyle { y_2-y_1 \over x_2-x_1}= { 12-4 \over 9-8}={8\over 1}=8.
10.
|
Find the slope of the
line passing through the points (–10, 10) and (0, 0).
|
Solution
Slope =\displaystyle { y_2-y_1 \over x_2-x_1}= { 10-0 \over -10-0}={10\over -10}=-1.
11.
|
Find the slope of the
graphed line.
|
12.
|
Find the slope of the
graphed line.
|
Slope =-{rise\over run}={3\over 9}={1\over 3}
[It is negative since it slopes downwards]
13.
|
If y varies
directly with x, and y = 110 when x = 100, find the
constant of variation k.
|
15.
|
The line graph below
shows the number of stray dogs in a certain city for the years listed.
(a) Which year had the least amount of stray
dogs
Solution: 1999
(b) Between which two years did the greatest
decrease in stray dogs occur?
|
16.
|
A student earns $0.65
for each mistake she finds in a text.
Sketch the equation of direct variation.
E=0.65m
|
17.
|
Find the slope and y-intercept.
8x + 5y = –50
|
5y = –50-8x
y = –50/5-8x/5
y=-1-{8\over 5}x
y=-{8\over 5}x-1
\therefore gradient m=-{8\over 5}, y intercept =-1.
18.
|
Write the equation of the line with slope 4 and y-intercept (0,
–5). Then graph the line.
|
y=mx+b
y=4x-5
19.
|
Write the equation of
the line with slope -{1\over 2} and y-intercept
(0, 3). Then graph the line.
|
y=-0.5x+3
20.
|
A line passing through
(10, 4) and (–3, y) is perpendicular to a line with slope -{13\over 14}. Find the value of y.
The perpendicular line has gradient which is begative reciprocal of -{13\over 14}, and therefore equals {14\over 13}.
{y-4\over -3-10}={14\over 13}
y-4=-14
y=-10
|
21.
|
An airplane covered 15
miles of its route while decreasing its altitude by 31,000 feet. Find the slope of the airplane's line of
descent. Round to the nearest
hundredth. [Hint: 1 mi = 5280 feet.]
|
22.
|
Determine which two
equations represent parallel lines.
|
(b) and (c) since the number in front of x is the same (in gradient intercept form)
23.
|
Determine which two
equations represent perpendicular lines.
(a) y=9x-9 (b) y={1\over 9}x+9 (c) y=-{1\over 9}x+9 (d) y={1\over 9}x-9 solution The theory says that if gradients are perpendicular then their product is -1. (a) and (c) since the product of the gradients equals -1, that is 9\times -{1\over 9}=-1 |
24.
|
Write the equation of
the line that passes through point (0, –8) with a slope of a {4\over 5}.
|
y={4\over 5}x-8
5y=4x-40
4x-5y-40=0
25.
|
Write the equation of
the line passing through (6, 37) and (1, 12).
gradient is m={37-12\over 6-1}={25\over 5}=5 y-y_1=m(x-x_1) y-12=5(x-1) y-12=5x-5 y=5x+7 or 5x-y+7=0. |
No comments:
Post a Comment