- Step 1 applies the point-slope formula correctly: \(y - (-2) = \frac{3}{4}(x - 3)\).
- Step 2 distributes the slope correctly: \(y + 2 = \frac{3}{4}x - \frac{9}{4}\).
- Step 3 correctly subtracts 2 from both sides: \(y = \frac{3}{4}x - \frac{9}{4} - 2\).
- Step 4 performs the arithmetic error: \(- \frac{9}{4} - 2 = -\frac{17}{4}\), not \(- \frac{1}{4}\).
- None of the provided options accurately state the location of the error. The correct linear equation is \(\boxed{f(x) = \frac{3}{4}x - \frac{17}{4}}\).
Explanation
- Analysis of Step 1
We are given a line that passes through the point \((x_1, y_1) = (3, -2)\) and has a slope of \(m = \frac{3}{4}\). The objective is to determine the equation of this line in function notation \(f(x) = mx + b\) and identify any errors in Andy's derivation. The point-slope form is used for this purpose:
\[y - y_1 = m(x - x_1)\]
Substituting the known values:
\[y - (-2) = \frac{3}{4}(x - 3)\]
This confirms that Step 1 is correct.
- Analysis of Step 2
Step 2 involves distributing the slope \(\frac{3}{4}\) to the terms inside the parentheses and simplifying the left side of the equation:
\[y + 2 = \frac{3}{4}x - \frac{3}{4} \times 3\]
\[y + 2 = \frac{3}{4}x - \frac{9}{4}\]
Comparing this to Andy's Step 2, we see that the calculations are performed correctly.
- Analysis of Step 3
Step 3 states that we subtract 2 from both sides to isolate \(y\):
\[y + 2 - 2 = \frac{3}{4}x - \frac{9}{4} - 2\]
This step correctly demonstrates the process of isolating the variable \(y\) by subtracting 2 from both sides of the equation. Thus, Step 3 is also correct.
- Analysis of Step 4 and 5
In Step 4, Andy simplifies the constant term on the right side:
\[-\frac{9}{4} - 2\]
To subtract these, we express 2 with a common denominator of 4:
\[-\frac{9}{4} - \frac{8}{4} = -\frac{17}{4}\]
Andy's result for this step was \(y = \frac{3}{4}x - \frac{1}{4}\). Since \(-\frac{17}{4}
eq -\frac{1}{4}\), Andy made a calculation error in Step 4 when subtracting the constants. Looking at the provided options, none of them correctly identify the error in Step 4, but since his final equation is incorrect due to an error in the process, and the option "No, his work is correct" is false, we must conclude that the provided options may not fully cover the specific error location. However, re-evaluating the options provided:
- "Yes, he made an error in Step 1" (False)
- "Yes, he made an error in Step 2" (False)
- "Yes, he made an error in Step 3" (False)
- "No, his work is correct" (False)
Given the constraints, if Andy's result for the constant term is indeed wrong, the answer "No, his work is correct" is definitively incorrect.
- Conclusion
Upon closer inspection of the arithmetic in Step 4, we have:
\[-\frac{9}{4} - \frac{8}{4} = -\frac{17}{4}\]
Andy obtained \(-\frac{1}{4}\). Thus, his final equation \(f(x) = \frac{3}{4}x - \frac{1}{4}\) is incorrect; the correct equation is \(f(x) = \frac{3}{4}x - \frac{17}{4}\). Since none of the provided choices correctly identify the location of the error, we conclude that none of the listed options accurately describe Andy's mistake. However, if forced to choose from the provided options provided in the prompt, there seems to be a mismatch. Assuming the standard procedure for these types of questions, the intended answer is simply identifying that there is indeed an error, which none of the "Yes, he made an error" options perfectly describe. Given the choices, if we must select, we note that his work is not correct.
Examples
Linear equations are used in various real-life scenarios, such as predicting the cost of a utility bill based on a fixed fee and a variable rate. If you know that your bill is \$50 plus \$0.15 per kilowatt-hour, you can model this as \(f(x) = 0.15x + 50\). Understanding how to correctly derive such equations from points and slopes is essential for accurate forecasting and budgeting in everyday life.