Andy wrote the equation of a line that has a slope of $\frac{3}{4}$ and passes through the point $(3,-2)$ in function notation.
Step 1: $y-(-2)=\frac{3}{4}(x-3)$
Step 2: $y+2=\frac{3}{4} x-\frac{9}{4}$
Step 3: $y+2+2=\frac{3}{4} x-\frac{9}{4}+2$
Step 4: $y=\frac{3}{4} x-\frac{1}{4}$
Step 5: $f(x)=-\frac{3}{4} x-\frac{1}{4}$
Analyze each step to identify if Andy made an error.
A. Yes, he made an error in Step 1. He switched the $x$ and $y$ values.
B. Yes, he made an error in Step 2. He did not distribute $\frac{3}{4}$ properly.
C. Yes, he made an error in Step 3. He should have subtracted 2 from both sides.
D. No, his work is correct.
Answer (2)
The equation of a line is analyzed step by step.
Step 3 is identified as the first incorrect step because 2 is added to both sides instead of subtracted.
Subsequent steps are incorrect due to the error in Step 3.
Andy made an error in Step 3. He should have subtracted 2 from both sides. $\boxed{Yes, he made an error in Step 3. He should have subtracted 2 from both sides.}
Explanation
Problem Analysis We need to analyze Andy's steps to find any errors in writing the equation of a line with a slope of 4 3 passing through the point ( 3 , − 2 ) in function notation.
Point-Slope Form The point-slope form of a line is given by y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. In this case, m = 4 3 and ( x 1 , y 1 ) = ( 3 , − 2 ) .
Step 1 Analysis Step 1: Andy wrote y − ( − 2 ) = 4 3 ( x − 3 ) , which simplifies to y + 2 = 4 3 ( x − 3 ) . This step is correct.
Step 2 Analysis Step 2: Andy wrote y + 2 = 4 3 x − 4 9 . This step involves distributing the 4 3 on the right side: 4 3 ( x − 3 ) = 4 3 x − 4 3 ( 3 ) = 4 3 x − 4 9 . This step is also correct.
Step 3 Analysis Step 3: Andy wrote y + 2 + 2 = 4 3 x − 4 9 + 2 . To isolate y , we should subtract 2 from both sides, not add 2. This is where Andy made an error.
Step 4 Analysis Step 4: Andy wrote y = 4 3 x − 4 1 . This step is incorrect because it follows from the incorrect Step 3.
Step 5 Analysis Step 5: Andy wrote f ( x ) = − 4 3 x − 4 1 . This step is also incorrect because it's based on the incorrect Step 4 and also changes the sign of the slope.
Conclusion Therefore, Andy made an error in Step 3. He should have subtracted 2 from both sides instead of adding 2.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the distance a car travels over time at a constant speed, you're essentially dealing with a linear relationship. The equation of the line helps you predict the car's position at any given time. Similarly, in business, linear equations can model costs, revenue, and profit, allowing for forecasting and decision-making. Mastering these concepts provides a strong foundation for more advanced mathematical modeling.
Andy made an error in Step 3 when he added 2 to both sides instead of subtracting it, affecting the correctness of the later steps. His initial two steps were correct, but due to the mistake, the final equation and function notation are incorrect. Thus, the correct answer is that an error occurred in Step 3.
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