What is the equation of a line that is parallel to the line $2x+5y=10$ and passes through the point $(-5,1)$? Check all that apply.
A. $y=-\frac{2}{5}x-1$
B. $2x+5y=-5$
C. $y=-\frac{2}{5}x-3$
D. $2x+5y=-15$
E. $y-1=-\frac{2}{5}(x+5)$
Answer (2)
Find the slope of the given line 2 x + 5 y = 10 , which is − 5 2 .
Use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , with the point ( − 5 , 1 ) and slope − 5 2 to get y − 1 = − 5 2 ( x + 5 ) .
Convert the point-slope form to slope-intercept form: y = − 5 2 x − 1 .
Convert the slope-intercept form to standard form: 2 x + 5 y = − 5 .
The equations are: y − 1 = − 5 2 ( x + 5 ) , y = − 5 2 x − 1 , 2 x + 5 y = − 5 .
Explanation
Problem Analysis We are given a line 2 x + 5 y = 10 and a point ( − 5 , 1 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point.
Finding the Slope First, let's find the slope of the given line. We can rewrite the equation in slope-intercept form, y = m x + b , where m is the slope. So, 5 y = − 2 x + 10 , and y = − 5 2 x + 2 . The slope of the given line is − 5 2 .
Parallel Lines Have Equal Slopes Since the line we are looking for is parallel to the given line, it has the same slope. Therefore, the slope of the new line is also − 5 2 .
Using Point-Slope Form Now, we can use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point ( − 5 , 1 ) . Plugging in the values, we get y − 1 = − 5 2 ( x − ( − 5 )) , which simplifies to y − 1 = − 5 2 ( x + 5 ) . This matches one of the given options.
Converting to Slope-Intercept Form Let's convert the point-slope form to slope-intercept form: y − 1 = − 5 2 x − 2 . Adding 1 to both sides, we get y = − 5 2 x − 1 . This matches another one of the given options.
Converting to Standard Form Now, let's convert the slope-intercept form to standard form: y = − 5 2 x − 1 . Multiplying both sides by 5, we get 5 y = − 2 x − 5 . Adding 2 x to both sides, we get 2 x + 5 y = − 5 . This matches another one of the given options.
Final Answer Therefore, the equations of the line that are parallel to the line 2 x + 5 y = 10 and passes through the point ( − 5 , 1 ) are: y − 1 = − 5 2 ( x + 5 ) y = − 5 2 x − 1 2 x + 5 y = − 5
Examples
Understanding parallel lines is crucial in various real-world applications, such as designing roads or buildings. For instance, when architects design a building, they need to ensure that walls are parallel to each other for structural stability and aesthetic appeal. Similarly, civil engineers use the concept of parallel lines when designing roads to ensure that lanes run smoothly and safely alongside each other. The equation of a line and the concept of parallel lines are fundamental in coordinate geometry and have practical applications in various fields.
The equations of the line that is parallel to 2 x + 5 y = 10 and passes through the point ( − 5 , 1 ) are given by the equations in options A, B, and E. They can be expressed in different forms: slope-intercept and standard forms. The correct equations are y = − 5 2 x − 1 , 2 x + 5 y = − 5 , and y − 1 = − 5 2 ( x + 5 ) .
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