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 (2)
Verify that the point ( 1 , 7 ) lies on the line f ( x ) = 4 x + 3 .
Find the slope of the line f ( x ) = 4 x + 3 , which is 4 .
Check each equation to see if it has the same slope and passes through the point ( 1 , 7 ) .
The equation y − 7 = 4 ( x − 1 ) represents the same line, which is y = 4 x + 3 . The answer is y − 7 = 4 ( x − 1 ) .
y − 7 = 4 ( x − 1 )
Explanation
Understanding the Problem The problem states that Chin was shown a graph of a line that contains the point ( 1 , 7 ) and that he wrote the equation f ( x ) = 4 x + 3 to represent the line. We need to determine which of the given equations could also represent the same line.
Verifying the Point 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 ) does indeed lie on the line f ( x ) = 4 x + 3 .
Finding the Slope The slope of the line f ( x ) = 4 x + 3 is 4 , since the equation is in slope-intercept form y = m x + b , where m is the slope.
Checking the Equations Now, let's examine each of the given equations to see if they represent the same line (i.e., have the same slope and pass through the point ( 1 , 7 ) ).
Analyzing Equation 1 Equation 1: y − 7 = 3 ( x − 1 ) . This is in point-slope form, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. Here, the slope is 3 , which is different from the slope of the given line ( 4 ). Therefore, this equation does not represent the same line.
Analyzing Equation 2 Equation 2: y − 1 = 3 ( x − 7 ) . This equation has a slope of 3 , which is different from the slope of the given line ( 4 ). Therefore, this equation does not represent the same line.
Analyzing Equation 3 Equation 3: y − 7 = 4 ( x − 1 ) . This equation has a slope of 4 , which is the same as the slope of the given line. Also, it passes through the point ( 1 , 7 ) . Therefore, this equation could represent the same line. Let's convert it to slope-intercept form to confirm: y − 7 = 4 ( x − 1 ) y − 7 = 4 x − 4 y = 4 x + 3 This is the same as the given equation f ( x ) = 4 x + 3 .
Analyzing Equation 4 Equation 4: y − 1 = 4 ( x − 7 ) . This equation has a slope of 4 , which is the same as the slope of the given line. However, it passes through the point ( 7 , 1 ) , not ( 1 , 7 ) . Therefore, this equation does not represent the same line.
Final Answer Therefore, the equation that could represent the same line is y − 7 = 4 ( x − 1 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, linear equations can model supply and demand curves. If you know a point on the demand curve (e.g., price and quantity demanded at that price) and the slope of the curve (how much the quantity demanded changes with each unit change in price), you can determine the entire demand curve using the point-slope form of a linear equation. This helps economists predict how changes in price will affect the quantity demanded, which is essential for making informed decisions about pricing and production.
The only equation that represents the same line as f ( x ) = 4 x + 3 is option C: y − 7 = 4 ( x − 1 ) since it has the same slope of 4 and passes through the point (1, 7).
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