1. Are the lines [tex]y=-3x+4[/tex] and [tex]x-3y=9[/tex] parallel, perpendicular, or neither?
2. Find the equation of the line (in slope-intercept form) that is perpendicular to [tex]x-y=0[/tex] and passes through the point [tex](4,6)[/tex].
Answer (2)
Determine the slope of the first line: m 1 = − 3 .
Determine the slope of the second line by rewriting the equation in slope-intercept form: m 2 = 3 1 .
Check if the lines are perpendicular by verifying if m 1 m 2 = − 1 . Since − 3 × 3 1 = − 1 , the lines are perpendicular.
Find the equation of the line perpendicular to x − y = 0 and passing through ( 4 , 6 ) . The slope of the perpendicular line is m = − 1 , and the equation is y = − x + 10 .
Explanation
Find the slope of the first line First, let's analyze the relationship between the lines y = − 3 x + 4 and x − 3 y = 9 . To do this, we need to find their slopes. The first line is already in slope-intercept form, y = m x + b , where m is the slope. So, the slope of the first line, m 1 , is − 3 .
Find the slope of the second line Now, let's rewrite the second equation, x − 3 y = 9 , in slope-intercept form. We can isolate y as follows:
x − 3 y = 9 − 3 y = − x + 9 y = − 3 − x + 9 y = 3 1 x − 3
So, the slope of the second line, m 2 , is 3 1 .
Determine the relationship between the lines To determine if the lines are parallel, perpendicular, or neither, we compare their slopes. If the slopes are equal ( m 1 = m 2 ), the lines are parallel. If the product of the slopes is − 1 ( m 1 m 2 = − 1 ), the lines are perpendicular. Otherwise, they are neither.
In this case, m 1 = − 3 and m 2 = 3 1 . Let's check if their product is − 1 :
m 1 × m 2 = − 3 × 3 1 = − 1
Since the product of the slopes is − 1 , the lines are perpendicular.
Find the slope of the third line Next, we need to find the equation of the line that is perpendicular to x − y = 0 and passes through the point ( 4 , 6 ) . First, let's find the slope of the line x − y = 0 . Rewriting this in slope-intercept form, we get:
x − y = 0 y = x
So, the slope of this line, m 3 , is 1 .
Find the slope of the perpendicular line The slope of a line perpendicular to this line, m 4 , is the negative reciprocal of m 3 . Therefore:
m 4 = − m 3 1 = − 1 1 = − 1
Use the point-slope form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point ( 4 , 6 ) and m is the slope m 4 = − 1 . Plugging in these values, we get:
y − 6 = − 1 ( x − 4 )
Rewrite in slope-intercept form Finally, we rewrite this equation in slope-intercept form, y = m x + b :
y − 6 = − x + 4 y = − x + 4 + 6 y = − x + 10
So, the equation of the line perpendicular to x − y = 0 and passing through the point ( 4 , 6 ) is y = − x + 10 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. Similarly, in road construction, perpendicular intersections are carefully planned to ensure safe and efficient traffic flow. The principles of slope and perpendicularity are also used in creating accurate blueprints and structural designs.
The lines y = − 3 x + 4 and x − 3 y = 9 are perpendicular to each other because the product of their slopes is − 1 . The line perpendicular to x − y = 0 that passes through ( 4 , 6 ) is given by the equation y = − x + 10 .
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