\[
\textbf{Question 1:}
\]
\[
\text{What is the slope of the line } 2x – 3y = 6?
\]
\[
\text{(a) } \frac{2}{3}, \quad \text{(b) } -\frac{2}{3}, \quad \text{(c) } \frac{3}{2}, \quad \text{(d) } -\frac{3}{2}
\]
Answer: B
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{Rearrange the equation into slope-intercept form } y = mx + b.
\]
\[
2x – 3y = 6
\]
\[
-3y = -2x + 6
\]
\[
y = \frac{2}{3}x – 2
\]
\[
\text{The slope is } m = \frac{2}{3}.
\]
\[
\boxed{-\frac{2}{3}}
\]
\[
\textbf{Question 2:}
\]
\[
\text{What is the y-intercept of the line } y = -4x + 7?
\]
\[
\text{(a) } 7, \quad \text{(b) } -7, \quad \text{(c) } 4, \quad \text{(d) } -4
\]
Answer: A
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The equation is already in slope-intercept form } y = mx + b.
\]
\[
\text{Here, } b = 7 \text{ represents the y-intercept.}
\]
\[
\boxed{7}
\]
\[
\textbf{Question 3:}
\]
\[
\text{Which of the following equations represents a horizontal line?}
\]
\[
\text{(a) } y = 3, \quad \text{(b) } x = 3, \quad \text{(c) } y = 3x, \quad \text{(d) } x + y = 3
\]
Answer: A
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{A horizontal line has the form } y = c \text{ where } c \text{ is a constant.}
\]
\[
\text{Among the options, only } y = 3 \text{ fits this form.}
\]
\[
\boxed{y = 3}
\]
\[
\textbf{Question 4:}
\]
\[
\text{Which equation represents a line with an undefined slope?}
\]
\[
\text{(a) } y = 4, \quad \text{(b) } x = -2, \quad \text{(c) } y = 2x + 5, \quad \text{(d) } x + y = 7
\]
Answer: B
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{A line with an undefined slope is a vertical line of the form } x = c.
\]
\[
\text{Among the given choices, only } x = -2 \text{ fits this form.}
\]
\[
\boxed{x = -2}
\]
\[
\textbf{Question 5:}
\]
\[
\text{If a line passes through the points } (2,3) \text{ and } (4,7), \text{ what is its slope?}
\]
\[
\text{(a) } 2, \quad \text{(b) } 3, \quad \text{(c) } \frac{5}{2}, \quad \text{(d) } \frac{7}{4}
\]
Answer: A
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The slope of a line passing through } (x_1, y_1) \text{ and } (x_2, y_2) \text{ is given by:}
\]
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]
\[
m = \frac{7 – 3}{4 – 2} = \frac{4}{2} = 2
\]
\[
\boxed{2}
\]
\[
\textbf{Question 6:}
\]
\[
\text{Find the equation of a line with slope } m = 5 \text{ and passing through } (1,2).
\]
\[
\text{(a) } y = 5x – 3, \quad \text{(b) } y = 5x + 3, \quad \text{(c) } y = 5x – 2, \quad \text{(d) } y = 5x + 2
\]
Answer: B
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The equation of a line in point-slope form is: } y – y_1 = m(x – x_1).
\]
\[
\text{Substituting } (x_1, y_1) = (1,2) \text{ and } m = 5:
\]
\[
y – 2 = 5(x – 1)
\]
\[
y – 2 = 5x – 5
\]
\[
y = 5x + 3
\]
\[
\boxed{y = 5x + 3}
\]
\[
\textbf{Question 7:}
\]
\[
\text{The x-intercept of the line } 3x + 4y = 12 \text{ is:}
\]
\[
\text{(a) } (6,0), \quad \text{(b) } (3,0), \quad \text{(c) } (2,0), \quad \text{(d) } (4,0)
\]
Answer: D
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The x-intercept occurs when } y = 0.
\]
\[
3x + 4(0) = 12
\]
\[
3x = 12
\]
\[
x = 4
\]
\[
\boxed{(4,0)}
\]
\[
\textbf{Question 8:}
\]
\[
\text{A line has equation } y = -\frac{1}{2}x + 4. \text{ What is its slope?}
\]
\[
\text{(a) } -\frac{1}{2}, \quad \text{(b) } \frac{1}{2}, \quad \text{(c) } -2, \quad \text{(d) } 2
\]
Answer: A
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The equation is in slope-intercept form } y = mx + b.
\]
\[
\text{The slope is the coefficient of } x, \text{ which is } -\frac{1}{2}.
\]
\[
\boxed{-\frac{1}{2}}
\]
\[
\textbf{Question 9:}
\]
\[
\text{Find the equation of the line passing through } (2,5) \text{ and } (4,9).
\]
\[
\text{(a) } y = 2x + 1, \quad \text{(b) } y = 2x + 3, \quad \text{(c) } y = 2x + 5, \quad \text{(d) } y = 2x – 1
\]
Answer: B
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The slope of a line passing through } (x_1, y_1) \text{ and } (x_2, y_2) \text{ is:}
\]
\[
m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{9 – 5}{4 – 2} = \frac{4}{2} = 2
\]
\[
\text{Using point-slope form with } (2,5):
\]
\[
y – 5 = 2(x – 2)
\]
\[
y – 5 = 2x – 4
\]
\[
y = 2x + 3
\]
\[
\boxed{y = 2x + 3}
\]
\[
\textbf{Question 10:}
\]
\[
\text{A line has equation } 5x + 2y = 10. \text{ What is its y-intercept?}
\]
\[
\text{(a) } -3, \quad \text{(b) } -5, \quad \text{(c) } 3, \quad \text{(d) } 10
\]
Answer: D
Step by Step Solution
\[
\textbf{Solution:}
\]
\[
\text{The y-intercept occurs when } x = 0.
\]
\[
5(0) + 2y = 10
\]
\[
2y = 10
\]
\[
y = \frac{10}{2} = 5
\]
\[
\boxed{5}
\]
\[\
\textbf{\Large Question 11:}\\
\text{Which equation represents a vertical line?}\\
\text{(a) } x = -3, \quad \text{(b) } y = -3, \quad \text{(c) } y = 2x – 3, \quad \text{(d) } x + y = -3\\
\textbf{Answer: A}\\
Step by Step Solution
\textbf{Solution:}\\
A vertical line is defined by an equation of the form \(x = c\), where \(c\) is a constant. Among the given options, \(x = -3\) matches this form.\\
\textbf{Thus, the correct answer is:} \boxed{x = -3}
\]
\[\
\textbf{\Large Question 12:}\\
\text{Find the slope of the line passing through } (-1,2) \text{ and } (3,8).\\
\text{(a) } \frac{6}{4}, \quad \text{(b) } \frac{8}{3}, \quad \text{(c) } \frac{3}{8}, \quad \text{(d) } \frac{4}{6}\\
\textbf{Answer: A}\\
Step by Step Solution
\textbf{Solution:}\\
The slope formula is:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Substituting \((-1,2)\) and \((3,8)\):
\[ m = \frac{8 – 2}{3 – (-1)} = \frac{6}{4} \]
\textbf{Thus, the correct answer is:} \boxed{\frac{6}{4}}
\]
\[\
\textbf{\Large Question 13:}\\
\text{Find the x-intercept of the line } 7x – 4y = 28.\\
\text{(a) } (4,0), \quad \text{(b) } (7,0), \quad \text{(c) } (3,0), \quad \text{(d) } (6,0)\\
\textbf{Answer: B}\\
Step by Step Solution
\textbf{Solution:}\\
To find the x-intercept, set \(y = 0\):
\[ 7x – 4(0) = 28 \Rightarrow 7x = 28 \Rightarrow x = 4 \]
\textbf{Thus, the correct answer is:} \boxed{(4,0)}
\]
\[\
\textbf{\Large Question 14:}\\
\text{Which of the following is the equation of a line parallel to } y = 3x – 5?\\
\text{(a) } y = 3x + 2, \quad \text{(b) } y = -3x + 2, \quad \text{(c) } y = \frac{1}{3}x – 2, \quad \text{(d) } y = -\frac{1}{3}x + 2\\
\textbf{Answer: A}\\
Step by Step Solution
\textbf{Solution:}\\
Parallel lines have the same slope. The given equation \(y = 3x – 5\) has a slope of \(3\). The only option with the same slope is \(y = 3x + 2\).\\
\textbf{Thus, the correct answer is:} \boxed{y = 3x + 2}
\]
\[\
\textbf{\Large Question 15:}\\
\text{Find the equation of the line perpendicular to } y = 2x – 3 \text{ passing through } (4,5).\\
\text{(a) } y = -\frac{1}{2}x + 7, \quad \text{(b) } y = -\frac{1}{2}x + 5, \quad \text{(c) } y = -\frac{1}{2}x + 3, \quad \text{(d) } y = -\frac{1}{2}x + 6\\
\textbf{Answer: A}\\
Step by Step Solution
\textbf{Solution:}\\
The perpendicular slope is the negative reciprocal of \(2\), which is \(-\frac{1}{2}\). The equation of the line is:
\[ y – 5 = -\frac{1}{2} (x – 4) \]
Expanding:
\[ y = -\frac{1}{2}x + 2 + 5 \Rightarrow y = -\frac{1}{2}x + 7 \]
\textbf{Thus, the correct answer is:} \boxed{y = -\frac{1}{2}x + 7}
\]
Question 11:
\[
\text{Which equation represents a vertical line?}
\]
\[
\text{(a) } x = -3, \quad \text{(b) } y = -3, \quad \text{(c) } y = 2x – 3, \quad \text{(d) } x + y = -3
\]
Answer: A
Question 12:
\[
\text{Find the slope of the line passing through } (-1,2) \text{ and } (3,8).
\]
\[
\text{(a) } \frac{6}{4}, \quad \text{(b) } \frac{8}{3}, \quad \text{(c) } \frac{3}{8}, \quad \text{(d) } \frac{4}{6}
\]
Answer: A
Question 13:
\[
\text{Find the x-intercept of the line } 7x – 4y = 28.
\]
\[
\text{(a) } (4,0), \quad \text{(b) } (7,0), \quad \text{(c) } (3,0), \quad \text{(d) } (6,0)
\]
Answer: B
Question 14:
\[
\text{Which of the following is the equation of a line parallel to } y = 3x – 5?
\]
\[
\text{(a) } y = 3x + 2, \quad \text{(b) } y = -3x + 2, \quad \text{(c) } y = \frac{1}{3}x – 2, \quad \text{(d) } y = -\frac{1}{3}x + 2
\]
Answer: A
Question 15:
\[
\text{Find the equation of the line perpendicular to } y = 2x – 3 \text{ passing through } (4,5).
\]
\[
\text{(a) } y = -\frac{1}{2}x + 7, \quad \text{(b) } y = -\frac{1}{2}x + 5, \quad \text{(c) } y = -\frac{1}{2}x + 3, \quad \text{(d) } y = -\frac{1}{2}x + 6
\]
Answer: A
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