Anya graphed the line $(y-2)=3(x-1)$ on the coordinate plane. What is the slope of Anya's line?
A. -3
B. -1
C. 1
D. 3
Answer (2)
Expand the equation: y − 2 = 3 x − 3 .
Isolate y : y = 3 x − 1 .
Identify the slope from the slope-intercept form.
The slope of the line is 3 .
Explanation
Understanding the Problem The equation of the line is given as ( y − 2 ) = 3 ( x − 1 ) . We need to find the slope of this line. The slope-intercept form of a linear equation is y = m x + b , where m represents the slope and b represents the y-intercept. Our goal is to rewrite the given equation in this form to easily identify the slope.
Expanding the Equation First, we expand the right side of the equation:
y − 2 = 3 ( x − 1 )
y − 2 = 3 x − 3
Isolating y Next, we isolate y by adding 2 to both sides of the equation:
y − 2 + 2 = 3 x − 3 + 2
y = 3 x − 1
Identifying the Slope Now that the equation is in slope-intercept form, y = 3 x − 1 , we can identify the slope as the coefficient of x . In this case, the coefficient of x is 3. Therefore, the slope of the line is 3.
Final Answer The slope of Anya's line is 3.
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, if you are analyzing the steepness of a hill for a hiking trail, the slope tells you how much the altitude changes for every unit of horizontal distance. Similarly, in economics, the slope of a supply or demand curve can tell you how sensitive the quantity supplied or demanded is to changes in price. In construction, the slope of a roof is important for water runoff and structural integrity. The equation of Anya's line can be used to model various real-world scenarios where a linear relationship exists between two variables.
The slope of Anya's line, represented by the equation ( y − 2 ) = 3 ( x − 1 ) , is determined to be 3 after isolating y in slope-intercept form. Thus, the correct answer is D. 3.
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