﻿ Equation of a Line Worksheet | Problems & Solutions Equation of a Line Worksheet

Equation of a Line Worksheet
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1.
Which of the following equations does not represent a straight line? a. $y$ = - $\frac{1}{10}$$x$ + 9 b. $\mathrm{xy}$ = 9 c. 10$x$ - $y$ - 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.

2.
Which of the following represents the point-slope form of an equation of a line? a. $y$ = $\mathrm{mx}$ + $b$ b. $\frac{x}{a}$ + $\frac{y}{b}$ = 1 c. $y$ - $y$1 = $m$($x$ - $x$1) d. A$x$ + B$y$ = 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.

3.
Which of the following lines is perpendicular to the line $y$ = 8$x$ + 7? a. $y$ = - $x$ + 7 b. $y$ = - $\frac{1}{8}$$x$ c. $y$ = - 7$x$ + 8 d. $y$ = $\frac{1}{7}$$x$

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

 Equation Slope y = - 1 / 8xy = - 7x + 8y = - x + 7y = 1 / 7x - 1 / 8 - 7 - 11 / 7

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

4.
Determine whether the statement is true or false. The line whose equation is $x$ - 5$y$ + 27 = 0 passes through (- 2, 5) and has a slope $\frac{1}{5}$. 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.

5.
Which of the following two lines are perpendicular?
I. $y$ = 6$x$
II. $y$ = - 6$x$
III. $y$ = $\frac{1}{6}$$x$
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.

 Equation Slope y = 6xy = - 6xy = 1 / 6xy = - x 6 - 61 / 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.

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).

7.
Find the equation of the line perpendicular to 5$x$ - 4$y$ = 9 and having the same $y$ - intercept. a. 5$x$ - 4$y$ + 9 = 0 b. 16$x$ - 20$y$ + 45 = 0 c. 16$x$ + 20$y$ + 45 = 0 d. 36$x$ + 45$y$ + 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.]

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

9.
Which of the following lines are parallel?
I. $y$ = $\frac{1}{2}$$x$
II. $y$ = $\frac{1}{5}$$x$ + 2
III. $y$ = $\frac{1}{2}$$x$ + $\frac{1}{5}$
IV. $y$ = - 5$x$ + $\frac{1}{2}$ 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.

 Equation Slope y = 1 / 2x y = 1 / 5x + 2y = 1 / 2x + 1 / 5y = - 5x + 1 / 2 1 / 21 / 51 / 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.

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