Find a general form of an equation of the line through the point $A$ that satisfies the given condition.
$A(9,8) ;$ perpendicular to the line $7 x+6 y=11$
Answer (2)
Find the slope of the given line 7 x + 6 y = 11 , which is − 6 7 .
Determine the slope of the perpendicular line, which is the negative reciprocal of the given line's slope, resulting in 7 6 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with point A ( 9 , 8 ) and the perpendicular slope to get y − 8 = 7 6 ( x − 9 ) .
Convert the equation to general form A x + B y = C , which simplifies to 6 x − 7 y = 2 .
Explanation
Problem Analysis We are given a point A ( 9 , 8 ) and a line 7 x + 6 y = 11 . Our goal is to find the equation of a line that passes through point A and is perpendicular to the given line.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with 7 x + 6 y = 11 , we isolate y :
6 y = − 7 x + 11
y = − 6 7 x + 6 11
So, the slope of the given line is − 6 7 .
Finding the Slope of the Perpendicular Line Next, we need to find the slope of the line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
m ⊥ = − ( − 6 7 ) 1 = 7 6
Using Point-Slope Form Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find the equation of the line passing through point A ( 9 , 8 ) . The point-slope form is:
y − y 1 = m ( x − x 1 )
Plugging in the values, we get:
y − 8 = 7 6 ( x − 9 )
Converting to General Form Finally, we need to convert the equation to the general form A x + B y = C . Starting from y − 8 = 7 6 ( x − 9 ) , we multiply both sides by 7 to eliminate the fraction:
7 ( y − 8 ) = 6 ( x − 9 )
7 y − 56 = 6 x − 54
Now, rearrange the terms to get the general form:
6 x − 7 y = − 54 + 56
6 x − 7 y = 2
Final Answer Thus, the general form of the equation of the line is 6 x − 7 y = 2 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For instance, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. This problem demonstrates how to find the equation of a line perpendicular to another, which can be applied to ensure structural integrity and precise angles in building designs. By using the principles of slopes and point-slope form, architects and engineers can accurately calculate and implement these perpendicular relationships in real-world applications.
To find the line through point A(9, 8) that is perpendicular to the line 7x + 6y = 11, we first find the slope of the given line as -7/6, then use the negative reciprocal to find the slope of the perpendicular line as 6/7. Using the point-slope form and rearranging gives us the equation in general form: 6x - 7y = 2.
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