Select the equation for the line crossing the point $(1,4)$ and with slope $=2$.
A. $y-1=4(x-2)$
B. $y+1=2(x+4)$
Answer (2)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 1 , 4 ) and slope 2 into the point-slope form: y − 4 = 2 ( x − 1 ) .
Simplify the equation: y = 2 x + 2 .
The correct equation of the line is: y − 4 = 2 ( x − 1 ) .
Explanation
Understanding the Problem We are given a point ( 1 , 4 ) and a slope m = 2 . We need to find the equation of the line that passes through this point and has this slope. The point-slope form of a line is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope.
Applying Point-Slope Form Substitute the given point ( 1 , 4 ) and slope 2 into the point-slope form: y − 4 = 2 ( x − 1 )
Comparing with Given Options Now, let's compare this equation with the given options: Option A: y − 1 = 4 ( x − 2 ) Option B: y + 1 = 2 ( x + 4 ) The equation we derived, y − 4 = 2 ( x − 1 ) , does not match either of the given options directly. However, we made an error in the options. The correct equation should be: y − 4 = 2 ( x − 1 ) So, let's rewrite the options to match the correct form.
However, the question asks us to select the equation from the given options. Let's analyze the options more closely. The point-slope form is y − y 1 = m ( x − x 1 ) .
Option A: y − 1 = 4 ( x − 2 ) . This implies the point is ( 2 , 1 ) and the slope is 4 .
Option B: y + 1 = 2 ( x + 4 ) . This implies the point is ( − 4 , − 1 ) and the slope is 2 .
Neither of these options matches the given point ( 1 , 4 ) and slope 2 . There seems to be an error in the provided options.
Verifying the Equation Let's verify the point-slope form with the given point and slope: y − 4 = 2 ( x − 1 ) y − 4 = 2 x − 2 y = 2 x + 2 Now, let's check if the given point ( 1 , 4 ) satisfies this equation: 4 = 2 ( 1 ) + 2 4 = 4 The point satisfies the equation. Since neither of the options is correct, we will derive the correct equation and state that the options are incorrect. The correct equation is: y − 4 = 2 ( x − 1 )
Stating the Correct Equation Since the given options are incorrect, we will provide the correct equation based on the given information. The equation of the line passing through the point ( 1 , 4 ) with a slope of 2 is: y − 4 = 2 ( x − 1 )
Examples
Understanding the equation of a line is crucial in various real-world applications. For instance, consider a scenario where a taxi charges a fixed initial fee plus a rate per mile. If the initial fee is $2 and the rate per mile is 2 , t h e t o t a l cos t y for x mi l esc anb ere p rese n t e d b y t h ee q u a t i o n y = 2x + 2$. This equation helps in calculating the fare for any distance traveled, making it a practical application of linear equations in everyday life.
The equation for the line crossing the point ( 1 , 4 ) with slope 2 is y − 4 = 2 ( x − 1 ) . Neither of the provided options represents this correct equation. Therefore, the options given do not match the desired line equation.
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