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Friday, 4 December 2015

A Bunch of Questions on Linear Functions (and direct variation)


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?

Solution:
To be solutions the pair of numbers (x,y) must satisfy the equation when you substitute the values into it.

(6,1):    LHS=6+2\times 1=8=RHS    YES
(8,0):    LHS=8+2\times 0= 8 =RHS    YES
(4,-2):    LHS=4+2\times -2=4-4=0\ne RHS    NO
(-4,6):    LHS=-4+2\times 6=8=RHS    YES
  
3.
Complete the ordered pairs for the equation 2x + y = 10.
            (5,    ), (    , 10), (    , –2), (0,    )
 Solution
When x=5:  2\times 5+y=10\implies y=5, so (5,5).
 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 2xy = 4.
 Solution

6.
Graph using the intercept method: $x + 3y = 6$.
  Solution
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.
            $4x – 3y = 6$
  Solution:
$4x – 3y = 6$
$– 3y = 6-4x$
$3y = -6+4x$
$3y =4x-6$
$y ={4\over 3}x-2$

So y intercept is 2 and gradient is m={4\over 3}.



8.
Graph using the intercept method: 2x + y = 4.
  Solution
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.
  Solution: x=5

12.
Find the slope of the graphed line.
  Solution:
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.

 Solution:
 y=kx.
110=k\times 100
\therefore k={110\over 100}=1.1
Update the equation: \therefore y=1.1 x 

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?
           Solution: 2002-2003

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
Solution
Make y the subject and then read the gradient m, and y-intercept b from the equation y=mx+b.

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=mx+b
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.
(a) y = {5\over 4}x + 3

(b) y ={4\over 5}x + 7

(c) y = {4\over 5}x – 7

(d) y = –{5\over 4}x + 7

Solution
(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.



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