Question

alejandra correctly wrote the equation $y - 3 = \frac{1}{5}(x - 10)$ to represent a line that her teacher sketched. the teacher then changed the line so it had a slope of 2, but still went through the same point. which equation should alejandra write to represent the new line?\
\
$y - 2 = \frac{1}{5}(x - 10)$\
\
$y - 3 = \frac{1}{5}(x - 2)$\
\
$y - 6 = 2(x - 10)$\
\
$y - 3 = 2(x - 10)$

Explanation:

Step1: Recall point - slope form

The point - slope form of a linear equation is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.

Step2: Identify the point and new slope

From the original equation \(y - 3=\frac{1}{5}(x - 10)\), we can see that the line passes through the point \((x_1,y_1)=(10,3)\) (by comparing with the point - slope form \(y - y_1=m(x - x_1)\)). The new slope \(m = 2\) and the line still passes through the same point \((10,3)\).

Step3: Substitute into point - slope form

Substitute \(x_1 = 10\), \(y_1=3\) and \(m = 2\) into the point - slope form \(y - y_1=m(x - x_1)\). We get \(y-3 = 2(x - 10)\).

Answer:

\(y - 3=2(x - 10)\) (the fourth option)