The equation of a linear function in point-slope form is $y-y_1=m\left(x-x_1\right)$. Harold correctly wrote the equation $y=3(x-7)$ using a point and the slope. Which point did Harold use?
A. $(7,3)$
B. $(0,7)$
C. $(7,0)$
D. $(3,7)$
Answer (2)
Rewrite the given equation y = 3 ( x − 7 ) in point-slope form: y − 0 = 3 ( x − 7 ) .
Compare this with the general point-slope form y − y 1 = m ( x − x 1 ) .
Identify the coordinates of the point: x 1 = 7 and y 1 = 0 .
State the point used by Harold: ( 7 , 0 ) .
Explanation
Understanding the Problem The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) . Harold wrote the equation y = 3 ( x − 7 ) . We need to identify the point ( x 1 , y 1 ) that Harold used.
Objective We want to determine the point ( x 1 , y 1 ) used by Harold to write the equation y = 3 ( x − 7 ) .
Solution Plan First, rewrite Harold's equation in the point-slope form: y − 0 = 3 ( x − 7 ) . Comparing this equation with the general point-slope form y − y 1 = m ( x − x 1 ) , we can identify the values of x 1 and y 1 .
Finding the Point From the equation y − 0 = 3 ( x − 7 ) , we can see that x 1 = 7 and y 1 = 0 . Therefore, the point Harold used is ( 7 , 0 ) .
Final Answer The point Harold used is ( 7 , 0 ) .
Examples
Understanding point-slope form is useful in many real-world scenarios. For example, if you know the rate at which you are saving money (the slope) and your current savings (a point), you can predict your future savings using this form. Similarly, in physics, if you know the velocity of an object (the slope) and its position at a certain time (a point), you can determine its position at any other time.
Harold used the point (7, 0) to write the equation y = 3 ( x − 7 ) . This was determined by rewriting the equation in point-slope form and identifying the coordinates. Therefore, the answer is ( 7 , 0 ) .
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