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?
$y-7=3(x-1)$
$y-1=3(x-7)$
$y-7=4(x-1)$
$y-1=4(x-7)$
Answer (2)
Verify that the point (1,7) satisfies the equation f ( x ) = 4 x + 3 .
Determine the slope of the line f ( x ) = 4 x + 3 , which is 4.
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point (1,7) and slope 4 to get y − 7 = 4 ( x − 1 ) .
Compare the derived equation with the given options and identify the matching equation: y − 7 = 4 ( x − 1 ) .
Explanation
Verifying the given point We are given that the line passes through the point (1,7) and is represented by the equation f ( x ) = 4 x + 3 . We need to find an equivalent equation from the given options. First, let's verify that the point (1,7) satisfies the equation f ( x ) = 4 x + 3 . Substituting x = 1 into the equation, we get f ( 1 ) = 4 ( 1 ) + 3 = 4 + 3 = 7 . So, the point (1,7) indeed lies on the line.
Finding the slope 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. The given equation is f ( x ) = 4 x + 3 , which can be written as y = 4 x + 3 . The slope of this line is m = 4 .
Using point-slope form Now, we can substitute the point (1,7) and the slope m = 4 into the point-slope form: y − 7 = 4 ( x − 1 ) . This is one of the given options.
Checking other options Let's check the other options:
y − 7 = 3 ( x − 1 ) : This equation has a slope of 3, which is different from the slope of the given line (which is 4). So, this is not the correct equation.
y − 1 = 3 ( x − 7 ) : This equation has a slope of 3, which is different from the slope of the given line (which is 4). Also, if we plug in x=1, we get y − 1 = 3 ( 1 − 7 ) = − 18 , so y = − 17 , which is not 7. So, this is not the correct equation.
y − 1 = 4 ( x − 7 ) : This equation has a slope of 4, which is the same as the slope of the given line. However, if we plug in x=1, we get y − 1 = 4 ( 1 − 7 ) = − 24 , so y = − 23 , which is not 7. So, this is not the correct equation.
Final Answer Therefore, the equation that represents the same line is y − 7 = 4 ( x − 1 ) .
Examples
Understanding linear equations and their various forms (slope-intercept, point-slope) is crucial in many 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 $3 and the rate per mile is 4 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb ere p rese n t e d a s y = 4x + 3$. If you know the cost for a particular distance (e.g., $7 for 1 mile), you can use the point-slope form to determine the equation of the line and predict the cost for any other distance. This concept extends to various fields like economics, physics, and engineering, where linear relationships are used to model and analyze data.
The equation that represents the same line as f ( x ) = 4 x + 3 and passes through the point ( 1 , 7 ) is y − 7 = 4 ( x − 1 ) . This matches one of the provided options. Therefore, the correct choice is y − 7 = 4 ( x − 1 ) .
;
