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Andy wrote the equation of a line that has a slope of [tex] \frac{3}{4} [/tex] and passes through the point [tex] (3, -2) [/tex] in function notation.

Step 1: [tex] y - (-2) = \frac{3}{4} (x - 3) [/tex]

Step 2: [tex] y + 2 = \frac{3}{4} x - \frac{9}{4} [/tex]

Step 3: [tex] y + 2 - 2 = \frac{3}{4} x - \frac{9}{4} - 2 [/tex]

Step 4: [tex] y = \frac{3}{4} x - \frac{17}{4} [/tex]

Step 5: [tex] f(x) = \frac{3}{4} x - \frac{17}{4} [/tex]

Analyze each step to identify if Andy made an error.

A. Yes, he made an error in Step 1. He switched the [tex] x [/tex] and [tex] y [/tex] values.
B. Yes, he made an error in Step 2. He did not distribute [tex] \frac{3}{4} [/tex] properly.
C. Yes, he made an error in Step 3. He should have subtracted 2 from both sides.
D. No, his work is correct.

Sagot :

GinnyAnswer
To analyze each step Andy took in writing the equation of the line, let's break down the steps one by one.

Step 1: [tex]\( y - (-2) = \frac{3}{4} (x - 3) \)[/tex]

In this step, Andy used the point-slope form of a linear equation, which is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is a point on the line.

Given:
- Slope, [tex]\( m = \frac{3}{4} \)[/tex]
- Point, [tex]\( (x_1, y_1) = (3, -2) \)[/tex]

Using the point-slope form:
[tex]\[ y - (-2) = \frac{3}{4}(x - 3) \][/tex]
This simplifies to:
[tex]\[ y + 2 = \frac{3}{4}(x - 3) \][/tex]

Step 1 appears correct because he correctly used the point-slope form equation without switching the x and y values.

Step 2: [tex]\( y + 2 = \frac{3}{4} x - \frac{9}{4} \)[/tex]

Next, he distributed [tex]\( \frac{3}{4} \)[/tex] to both terms inside the parenthesis:
[tex]\[ \frac{3}{4}(x - 3) = \frac{3}{4} x - \frac{3}{4} \times 3 \][/tex]
[tex]\[ = \frac{3}{4} x - \frac{9}{4} \][/tex]

Therefore:
[tex]\[ y + 2 = \frac{3}{4} x - \frac{9}{4} \][/tex]

This shows correct distribution of the slope, so Step 2 is correct.

Step 3: [tex]\( y + 2 + 2 = \frac{3}{4} x - \frac{9}{4} + 2 \)[/tex]

Andy added 2 to both sides of the equation:
[tex]\[ y + 2 = \frac{3}{4} x - \frac{9}{4} \][/tex]
[tex]\[ y + 2 - 2 = \frac{3}{4} x - \frac{9}{4} - 2 \][/tex]

Thus correcting would be simple subtraction:
[tex]\[ y = \frac{3}{4} x - \frac{9}{4} - 2 \][/tex]
Transforming:
[tex]\[ y = \frac{3}{4} x - \frac{17}{4} \][/tex]

He added instead of subtracted in this step - thus there was an error.

Step 4: [tex]\( y = \frac{3}{4} x - \frac{1}{4} \)[/tex]

Incorrect stage result based on prior error fixed that:

Yet shall show as [tex]\( y\equiv\frac{3}{4} x-\frac{17}{4}\)[/tex]

Step 5: [tex]\( f(x) = \frac{3}{4} x - \frac{1}{4} \)[/tex]

From Step 4 transitioning to function notation shows:
[tex]\[f(x)=\frac{3}{4} x-\frac{17}{4}\][/tex]

From correction of step 3 noticing correct concludes:
Errors found
Fix noting step progression showing prior:
Thus modifying steps accordingly reality:
Hence final correct transforming steps All proper.
Thus final presenting itself equal to stage being:
Thus concluding correctness process.
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