Chin was shown the graph of a line that contained point $(1,7)$. He wrote $f(x)=4 x+3$ to correctly represent the line. Which of these equations could represent the same line?
A. $y-7=3(x-1)$
B. $y-1=3(x-7)$
C. $y-7=4(x-1)$
D. $y-1=4(x-7)$
Answer (1)
Verify that the point ( 1 , 7 ) lies on the line f ( x ) = 4 x + 3 .
Find the point-slope form of the line using the point ( 1 , 7 ) and the slope 4 : y − 7 = 4 ( x − 1 ) .
Compare the point-slope form with the given options.
The equation that represents the same line is y − 7 = 4 ( x − 1 ) .
Explanation
Verify the point on the line First, let's verify that the point ( 1 , 7 ) lies on the line f ( x ) = 4 x + 3 . We substitute x = 1 into the equation: f ( 1 ) = 4 ( 1 ) + 3 = 4 + 3 = 7 Since f ( 1 ) = 7 , the point ( 1 , 7 ) indeed lies on the line f ( x ) = 4 x + 3 .
Find the point-slope form Now, let's analyze the given options to see which equation represents the same line. The equation of a line in point-slope form 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 of the line. We know that the line passes through the point ( 1 , 7 ) . The slope of the line f ( x ) = 4 x + 3 is 4 . Therefore, the equation of the line in point-slope form is: y − 7 = 4 ( x − 1 )
Compare with the options Now, let's compare this equation with the given options:
Option 1: y − 7 = 3 ( x − 1 ) . The slope is 3, which is incorrect. Option 2: y − 1 = 3 ( x − 7 ) . The point is ( 7 , 1 ) and the slope is 3, which is incorrect. Option 3: y − 7 = 4 ( x − 1 ) . The point is ( 1 , 7 ) and the slope is 4, which is correct. Option 4: y − 1 = 4 ( x − 7 ) . The point is ( 7 , 1 ) and the slope is 4, which is incorrect.
Final Answer Therefore, the equation that represents the same line is y − 7 = 4 ( x − 1 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For example, in economics, a linear equation can represent a cost function, where the slope represents the variable cost per unit and the y-intercept represents the fixed costs. If a company knows its cost function and a point on the line, they can determine the equation of the line and make predictions about future costs. Similarly, in physics, linear equations can describe motion with constant velocity, where the slope represents the velocity and the y-intercept represents the initial position. By understanding the equation of the line, one can predict the position of an object at any given time.
