Question

whats the mistake?
fill in the table below:

• analyze each problem.
• study the incorrect solution given.
• describe in your own words the mistake that was made.
• give the correct answer.

problem #1:
the graph of a function is a line that passes through the coordinates (4, -2) and (-1, 8).

which is an equation in terms of x and y for this function?
a. $y = -\frac{3}{10}x$ b. $y = -2x + 6$
c. $y = -\frac{1}{2}x$ d. $y = -\frac{10}{3}x + 6$

incorrect solution:
find the slope:
$\frac{-2 - 8}{4 - 1} = \frac{-10}{3}$
since d is the only answer with a slope of -10/3, it must be correct.

explain the mistake:

correct answer:

Explanation:

Explain the Mistake:

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \) (or \( \frac{y_1 - y_2}{x_1 - x_2} \)). In the incorrect solution, when calculating the slope between \((4, - 2)\) and \((-1, 8)\), the denominator was calculated as \( 4 - 1 \) instead of \( 4-(-1) \). The correct denominator should be \( 4-(-1)=4 + 1=5 \), not \( 3 \). So the slope calculation was wrong.

Correct Answer:

First, calculate the correct slope:
Using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), let \((x_1,y_1)=(4,-2)\) and \((x_2,y_2)=(-1,8)\).
Then \( m=\frac{8-(-2)}{-1 - 4}=\frac{8 + 2}{-5}=\frac{10}{-5}=- 2 \).

Now, use the point - slope form \( y - y_1=m(x - x_1) \). Let's use the point \((4,-2)\) and \( m=-2 \).
\( y-(-2)=-2(x - 4) \)
\( y + 2=-2x+8 \)
Subtract 2 from both sides: \( y=-2x + 6 \)

So the correct answer is B. \( y=-2x + 6 \)

Answer:

Explain the Mistake:

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \) (or \( \frac{y_1 - y_2}{x_1 - x_2} \)). In the incorrect solution, when calculating the slope between \((4, - 2)\) and \((-1, 8)\), the denominator was calculated as \( 4 - 1 \) instead of \( 4-(-1) \). The correct denominator should be \( 4-(-1)=4 + 1=5 \), not \( 3 \). So the slope calculation was wrong.

Correct Answer:

First, calculate the correct slope:
Using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), let \((x_1,y_1)=(4,-2)\) and \((x_2,y_2)=(-1,8)\).
Then \( m=\frac{8-(-2)}{-1 - 4}=\frac{8 + 2}{-5}=\frac{10}{-5}=- 2 \).

Now, use the point - slope form \( y - y_1=m(x - x_1) \). Let's use the point \((4,-2)\) and \( m=-2 \).
\( y-(-2)=-2(x - 4) \)
\( y + 2=-2x+8 \)
Subtract 2 from both sides: \( y=-2x + 6 \)

So the correct answer is B. \( y=-2x + 6 \)