Answer :
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.
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.
