Which pair of equations represents parallel lines?
$\begin{array}{l}
-\frac{2}{3} x+y=12 \\
y=-\frac{3}{2} x-1
\end{array}$
$\begin{array}{l}
3 x+y=-8 \\
y=3 x-8
\end{array}$
$\begin{array}{l}
-2 x+y+2=0 \\
y=-\frac{1}{2} x-4
\end{array}$
$\begin{array}{l}
x+2 y=8 \\
-x-2 y=3
\end{array}$
Answer (2)
Rewrite each equation in slope-intercept form ( y = m x + b ).
Compare the slopes and y-intercepts of each pair of equations.
Identify the pair with the same slope but different y-intercepts.
The fourth pair of equations represents parallel lines: x + 2 y = 8 − x − 2 y = 3 .
Explanation
Understanding the Problem We are given four pairs of equations and need to determine which pair represents parallel lines. Parallel lines have the same slope but different y-intercepts. We need to rewrite each equation in slope-intercept form ( y = m x + b ) to identify the slope ( m ) and y-intercept ( b ).
Plan of Action Let's rewrite each pair of equations in slope-intercept form ( y = m x + b ) and compare their slopes and y-intercepts.
Analyzing the First Pair For the first pair of equations:
Equation 1: − 3 2 x + y = 12 can be rewritten as y = 3 2 x + 12 .
Equation 2: y = − 2 3 x − 1 .
The slopes are 3 2 and − 2 3 , respectively. Since the slopes are different, these lines are not parallel.
Analyzing the Second Pair For the second pair of equations:
Equation 1: 3 x + y = − 8 can be rewritten as y = − 3 x − 8 .
Equation 2: y = 3 x − 8 .
The slopes are − 3 and 3 , respectively. Since the slopes are different, these lines are not parallel.
Analyzing the Third Pair For the third pair of equations:
Equation 1: − 2 x + y + 2 = 0 can be rewritten as y = 2 x − 2 .
Equation 2: y = − 2 1 x − 4 .
The slopes are 2 and − 2 1 , respectively. Since the slopes are different, these lines are not parallel.
Analyzing the Fourth Pair For the fourth pair of equations:
Equation 1: x + 2 y = 8 can be rewritten as 2 y = − x + 8 , so y = − 2 1 x + 4 .
Equation 2: − x − 2 y = 3 can be rewritten as − 2 y = x + 3 , so y = − 2 1 x − 2 3 .
The slopes are both − 2 1 . The y-intercepts are 4 and − 2 3 , respectively. Since the slopes are the same and the y-intercepts are different, these lines are parallel.
Conclusion The fourth pair of equations, x + 2 y = 8 and − x − 2 y = 3 , represents parallel lines because they have the same slope ( − 2 1 ) but different y-intercepts ( 4 and − 2 3 ).
Examples
Understanding parallel lines is crucial in various real-world applications, such as designing roads and buildings. For example, architects use parallel lines to create symmetrical and balanced structures. City planners use parallel lines when designing road layouts to ensure traffic flows smoothly and efficiently. In computer graphics, parallel lines are used to create perspective and depth in images.
The fourth pair of equations, x + 2 y = 8 and − x − 2 y = 3 , represents parallel lines because they both have the same slope of − 2 1 but different y-intercepts. Therefore, the answer is the last pair of equations. Parallel lines are defined as lines that have the same slope but different y-intercepts.
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