Which equations represent the line that is parallel to [tex]$3 x-4 y=7$[/tex] and passes through the point [tex]$(-4,-2)$[/tex]? Select two options. [tex]$y=\frac{-3}{4} x+1$[/tex] [tex]$3 x-4 y=-4$[/tex] [tex]$4 x-3 y=-3$[/tex] [tex]$y-2=\frac{3}{4}(x-4)$[/tex] [tex]$y+2=\frac{3}{4}(x+4)$[/tex]
Answer (2)
Find the slope of the given line 3 x − 4 y = 7 , which is 4 3 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( − 4 , − 2 ) and slope 4 3 to get y + 2 = 4 3 ( x + 4 ) .
Simplify the point-slope form to slope-intercept form: y = 4 3 x + 1 .
Convert to standard form: 3 x − 4 y = − 4 . The two correct equations are y + 2 = 4 3 ( x + 4 ) and 3 x − 4 y = − 4 .
y + 2 = 4 3 ( x + 4 ) and 3 x − 4 y = − 4
Explanation
Problem Analysis We are given a line 3 x − 4 y = 7 and a point ( − 4 , − 2 ) . We need to find two equations that represent the line parallel to the given line and passing through the given point.
Finding the Slope First, let's find the slope of the given line. We can rewrite the equation 3 x − 4 y = 7 in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Solving for y , we get:
3 x − 4 y = 7 − 4 y = − 3 x + 7 y = − 4 − 3 x + − 4 7 y = 4 3 x − 4 7
Slope of Parallel Line The slope of the given line is 4 3 . Since parallel lines have the same slope, the slope of the line we are looking for is also 4 3 .
Point-Slope Form Now, we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point ( − 4 , − 2 ) . Substituting the values, we get:
y − ( − 2 ) = 4 3 ( x − ( − 4 )) y + 2 = 4 3 ( x + 4 )
First Matching Equation One of the options is y + 2 = 4 3 ( x + 4 ) , which matches our equation.
Slope-Intercept Form Let's simplify the point-slope form to get the slope-intercept form:
y + 2 = 4 3 ( x + 4 ) y + 2 = 4 3 x + 4 3 ( 4 ) y + 2 = 4 3 x + 3 y = 4 3 x + 3 − 2 y = 4 3 x + 1
Standard Form Now, let's convert this to the standard form A x + B y = C :
y = 4 3 x + 1 4 y = 3 x + 4 − 3 x + 4 y = 4 3 x − 4 y = − 4
Second Matching Equation Another option is 3 x − 4 y = − 4 , which matches our equation.
Final Answer Therefore, the two equations that represent the line parallel to 3 x − 4 y = 7 and passing through the point ( − 4 , − 2 ) are y + 2 = 4 3 ( x + 4 ) and 3 x − 4 y = − 4 .
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. If a designer needs to create a line parallel to an existing wall and passing through a specific point, they would use the principles we applied in this problem. Imagine a landscape architect designing a garden path parallel to a fence; they would use the same slope as the fence and ensure the path passes through a desired location, applying the point-slope form to map out the path accurately.
The equations representing the line parallel to 3 x − 4 y = 7 and passing through the point ( − 4 , − 2 ) are y + 2 = 4 3 ( x + 4 ) and 3 x − 4 y = − 4 .
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