Question

functions and linear equations

exercises - parallel and perpendicular lines

1
which of the following equations represents a line that is parallel to the line with equation \\(y = -\frac{1}{2}x + 5\\) and contains the point \\((-2, \frac{1}{2})\\)?
a) \\(x - 2y = -3\\)
b) \\(x + 2y = -1\\)
c) \\(2x - y = -5\\)
d) \\(2x + y = -3\\)

2
which of the following equations represents a line that passes through \\((7, 6)\\) and is parallel to the \\(x\\)-axis?
a) \\(x = 6\\)
b) \\(y = 7\\)
c) \\(y = 7\\)
d) \\(y = 6\\)

3
which of the following equations represents a line that passes through \\((-5, 1)\\) and is parallel to the \\(y\\)-axis?
a) \\(y = -5\\)
b) \\(y = 1\\)
c) \\(x = -5\\)
d) \\(x = 1\\)

4
a line passes through the points \\((-1, 2)\\) and \\((5, b)\\), and is parallel to the graph of the equation \\(4x - 2y = 13\\). what is the value of \\(b\\)?

5
in the \\(xy\\)-plane above, line \\(\ell\\) is parallel to line \\(m\\). what is the value of \\(b\\)?

6
in the \\(xy\\)-plane above, if line \\(\ell\\) is perpendicular to line \\(t\\), what is the value of \\(a\\)?

Explanation:

⚡ Using what you learned: parallel and perpendicular lines_equations

Step 1: Find the slope of the given line

The given equation is:

$$ 4x - 2y = 13 $$

Convert to slope-intercept form \( y = mx + c \):

$$ -2y = -4x + 13 $$

$$ y = 2x - \frac{13}{2} $$

The slope \( m \) of the given line is \( 2 \). Since the new line is parallel, its slope is also \( m = 2 \).

Step 2: Set up the slope equation with the given points

The parallel line passes through \( (-1, 2) \) and \( (5, b) \). Use the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

$$ 2 = \frac{b - 2}{5 - (-1)} $$

$$ 2 = \frac{b - 2}{6} $$

Step 3: Solve for b

$$ 2 \cdot 6 = b - 2 $$

$$ 12 = b - 2 $$

$$ b = 14 $$

Answer:

14