Select the correct answer from each drop-down menu.
Consider the given equation.
[tex]$3 x+2 y=8$[/tex]
The equation [tex]$y=$[/tex] [blank] [tex]$x+$[/tex] [blank] represents the line parallel to the given equation and passes through the point [tex]$(-2,5)$[/tex].
Answer (2)
Rewrite the given equation in slope-intercept form to find its slope: y = − 2 3 x + 4 .
The slope of the parallel line is the same as the slope of the given line: m = − 2 3 .
Use the point-slope form with the point ( − 2 , 5 ) to find the y-intercept: 5 = − 2 3 ( − 2 ) + b , which gives b = 2 .
The equation of the parallel line is y = − 2 3 x + 2 , so the answer is y = − 2 3 x + 2 .
Explanation
Understanding the Problem We are given the equation 3 x + 2 y = 8 and asked to find the equation of a line parallel to it that passes through the point ( − 2 , 5 ) . The equation should be in the slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope First, we need to find the slope of the given line. To do this, we rewrite the equation 3 x + 2 y = 8 in slope-intercept form.
Slope-Intercept Form of the Given Line Solving for y , we get:
2 y = − 3 x + 8
y = − 2 3 x + 4
So, the slope of the given line is − 2 3 .
Determining the Slope of the Parallel Line Since parallel lines have the same slope, the slope of the line we are looking for is also − 2 3 . Now we know that the equation of the line we want is of the form y = − 2 3 x + b .
Finding the Y-Intercept We are given that the line passes through the point ( − 2 , 5 ) . We can use this point to find the y-intercept, b , by substituting the coordinates of the point into the equation y = − 2 3 x + b .
Calculating the Y-Intercept Substituting x = − 2 and y = 5 into the equation, we get:
5 = − 2 3 ( − 2 ) + b
5 = 3 + b
b = 5 − 3 = 2
The Equation of the Parallel Line Now we have the slope m = − 2 3 and the y-intercept b = 2 . Therefore, the equation of the line parallel to 3 x + 2 y = 8 and passing through the point ( − 2 , 5 ) is y = − 2 3 x + 2 .
Examples
Understanding parallel lines is crucial in various real-world applications, such as architecture and urban planning. For instance, when designing a building, architects ensure that walls are parallel to each other for structural stability and aesthetic appeal. Similarly, city planners use the concept of parallel lines to design roads and streets that run alongside each other without intersecting, optimizing traffic flow and minimizing congestion. The equation of a line and its parallel counterpart helps in creating precise and efficient designs in these fields.
The equation of the line parallel to 3 x + 2 y = 8 and passing through the point ( − 2 , 5 ) is y = − 2 3 x + 2 . This is established by using the slope of the original line and the point-slope form of the line equation. The slope remains the same for parallel lines.
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