Question

Carlota wrote the equation y + 1 = 2 (x - 3) for the line passing through the points (-1, 3) and (2, 9). Explain and correct her error.

Answer 1

For this case we have the following equation:

$y + 1 = 2 (x - 3)$

That can be rewritten in the form $y = mx + b$

Where:

• m is the slope of the line

• b is the cut point

So, we have:

$y + 1 = 2 (x - 3)\\y + 1 = 2x-6\\y = 2x-6-1\\y = 2x-7$

Where:

$m = 2$ is the slope

$b = -7$ is the cut point

Carlota has the following points:

(-1, 3) and (2, 9)

To know if the line $y = 2x-7$ passes through these points, we must replace them in the equation and the equality must be fulfilled. So:

Point (-1, 3):

Substituting:

$3 = 2 (-1) -7\\3 = -2-7$

$3 = -9$ It's false, equality is not met. The point (-1, 3) does not go through the line.

The equation written by Carlota is erroneous, the procedure to follow is:

Given$(x1, y1) = (- 1, 3)$ and$(x2, y2) = (2, 9)$, we find the slope:

$m=\frac{(y2-y1)}{(x2-x1)}$

$m=\frac{(9-3)}{(2-(-1))}$

$m=\frac{6}{3}$

$m=2$

We observe that the slope found by Carlota is the same. Let's see cut point "b". For this we substitute any of the points given in the equation:

$y = 2x + b$

Substituting (2,9) we have:

$9 = 2 (2) + b\\9 = 4 + b\\b = 9-4\\b = 5$

Thus, Carlota's error was at the cut-off point. The correct equation of the line that passes through the given points is $y = 2x + 5$

Answer:

The correct equation of the line that passes through the given points is $y = 2x + 5$

Carlota's mistake was at the cutoff point