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Equation of a Line Worksheet

Equation of a Line Worksheet
  • Page 1
 1.  
Which of the following equations does not represent a straight line?
a.
y = - 1 10x + 9
b.
xy = 9
c.
10x - y - 8 = 0
d.
(y - 9) = 10(x - 8)


Solution:

y = - 1 / 10 x + 9 is a straight line.
[Equation of a straight line in slope-intercept form.]

(y - 9) = 10 (x - 8) is a straight line.
[Equation of a straight line in point-slope form.]

10x - y - 8 = 0 is a straight line.
[Equation of a straight line in standard form.]

xy = 9 is not a straight line, because this cannot be written as any one of the above forms of a line.


Correct answer : (2)
 2.  
Which of the following represents the point-slope form of an equation of a line?
a.
y = mx + b
b.
xa + yb = 1
c.
y - y1 = m(x - x1)
d.
Ax + By = C


Solution:

The point-slope form of an equation of a line is y - y1 = m (x - x1) where m is the slope and (x1, y1) are the coordinates of a given point on the line.


Correct answer : (3)
 3.  
Which of the following lines is perpendicular to the line y = 8x + 7?
a.
y = - x + 7
b.
y = - 1 8x
c.
y = - 7x + 8
d.
y = 1 7x


Solution:

Two lines are perpendicular if the product of their slopes is - 1.

The slope of the line y = 8x + 7 is 8.
[Find the slope of the line using y = mx + b.]

If x is the slope of the line perpendicular to the line y = 8x + 7, then m × 8 = - 1, m = - 1 / 8

EquationSlope
y = - 1 / 8x
y = - 7x + 8
y = - x + 7
y = 1 / 7x
- 1 / 8
- 7
- 1
1 / 7


So, the line y = - 1 / 8x is perpendicular to the line y = 8x + 7.


Correct answer : (2)
 4.  
Determine whether the statement is true or false. The line whose equation is x - 5y + 27 = 0 passes through (- 2, 5) and has a slope 1 5.
a.
False
b.
True


Solution:

x - 5y + 27 = 0
[Equation of the line.]

y = x5 + 27 / 5
[Slope-intercept form of the line.]

Slope = 1 / 5
[Slope of y = mx + c is m.]

Substitute (- 2, 5) in x - 5y + 27 = 0.

- 2 - 5 (5) + 27 = 0

0 = 0, which is always true.

So, the given statement is true.


Correct answer : (2)
 5.  
Which of the following two lines are perpendicular?
I. y = 6x
II. y = - 6x
III. y = 16x
IV. y = - x
a.
I and II
b.
II and IV
c.
II and III
d.
III and I


Solution:

The slope of the line of the form y = mx is m.

EquationSlope
y = 6x
y = - 6x
y = 1 / 6x
y = - x
6
- 6
1 / 6
- 1


Two lines are perpendicular if the product of their slopes is - 1.

The product of the slopes of the lines y = - 6x and y = 16x gives - 1.

So, the lines II and III are perpendicular.


Correct answer : (3)
 6.  
Which of the following statements is true for the graph of the function y = 3?
a.
The graph of y = 3 in the coordinate plane is a single point (0, 3).
b.
The graph of y = 3 is a line parallel to the y-axis, at a distance of 3 units to the right of it.
c.
The graph of y = 3 is a line parallel to the x-axis, at a distance of 3 units above the x-axis.
d.
The graph of y = 3 is a line parallel to the x-axis, at a distance of 3 units below the x-axis.


Solution:


The graph of y = 3 is a line parallel to the x - axis, at a distance of 3 units above the x - axis.

So, the graph of y = 3 is not a single point (0, 3).


Correct answer : (3)
 7.  
Find the equation of the line perpendicular to 5x - 4y = 9 and having the same y - intercept.
a.
5x - 4y + 9 = 0
b.
16x - 20y + 45 = 0
c.
16x + 20y + 45 = 0
d.
36x + 45y + 20 = 0


Solution:

5x - 4y = 9
[Equation of a line.]

y = 5 / 4x - 9 / 4
[Rewrite it in y = mx + c form.]

So, the slope, m = 5 / 4 and y-intercept, c = - 9 / 4

Slope of the perpendicular line = - 1m = - 4 / 5

y - intercept of the required line = - 9 / 4

The equation of the perpendicular line is: y = - 4 / 5x - 9 / 4
[y = mx + c.]

20y = - 16x - 45
[Multiply both sides by 20.]

16x + 20y + 45 = 0
[Simplify.]


Correct answer : (3)
 8.  
Find the equation of the line passing through (1, 0) and parallel to the line passing through (0, 0) and (10, - 5).
a.
4x + 2y - 1 = 0
b.
x + 2y - 4 = 0
c.
x - 2y + 1 = 0
d.
x + 2y - 1 = 0


Solution:

The given line is passing through the points (0, 0) and (10, - 5).

Slope = y2-y1x2-x1 = -5-010-0 = - 1 / 2

Slope of the line parallel to the given line = - 1 / 2
[Parallel lines slopes are equal.]

So, equation of the parallel line is, y = - 1 / 2x + c.
[Slope-intercept form.]

This line passes through the point (1, 0).

0 = - 1 / 2(1) + c

c = 12
[Simplify.]

Therefore, the equation of the line is y = - 1 / 2x + 1 / 2.

2y = - x + 1, so x + 2y - 1 = 0
[Multiply both sides by 2.]


Correct answer : (4)
 9.  
Which of the following lines are parallel?
I. y = 1 2x
II. y = 1 5x + 2
III. y = 1 2x + 1 5
IV. y = - 5x + 1 2
a.
II and II
b.
I, II and III
c.
II, III and IV
d.
I and III


Solution:

Two lines are parallel if their slopes are equal.

Therefore, the two lines are parallel.

EquationSlope
y = 1 / 2x
y = 1 / 5x + 2
y = 1 / 2x + 1 / 5
y = - 5x + 1 / 2
1 / 2
1 / 5
1 / 2
- 5


Slopes of the lines y = 1 / 2x and y = 1 / 2x + 1 / 5 are equal.

So, the lines I and III are parallel.


Correct answer : (4)
 10.  
What is the equation of the line passing through the point (4, 5) and perpendicular to the line y = 3x + 8?
a.
x + 4 = - 3(y - 5)
b.
y - 5 = - 1 3(x - 4)
c.
x - 4 = 3(y - 5)
d.
y + 5 = - 1 3(x - 4)


Solution:

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

Slope of the line y = 3x + 8 is 3.
[Compare with the equation in step 1.]

Slope of the line perpendicular to y = 3x + 8 = - 1 / 3
[Product of the slopes of perpendicular lines is -1.]

The equation of the line passing through the point (x1, y1) with slope 'm' in point-slope form is y - y1 = m(x - x1).

Point (x1, y1) = (4, 5) and slope m of the perpendicular line = - 1 / 3 .

y - 5 = - 1 / 3(x - 4)
[Substitute x1 = 4, y1 = 5 and m = - 1 / 3 in the equation in step 4.]

The equation of the line passing through the point (4, 5) is y - 5 = - 1 / 3(x - 4).


Correct answer : (2)

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