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?\
\\(\circ\\ y - 6 = 2(x - 10)\\)\
\\(\circ\\ y - 2 = \frac{1}{5}(x - 10)\\)\
\\(\circ\\ y - 3 = \frac{1}{5}(x - 2)\\)\
\\(\circ\\ y - 3 = 2(x - 10)\\)

Explanation:

Step1: Recall Point - Slope Form

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$, 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 identify that the point $(x_1,y_1)$ is $(10,3)$ (by comparing with $y - y_1=m(x - x_1)$) and the original slope $m=\frac{1}{5}$. The teacher changed the slope to $m = 2$ but kept the same point $(10,3)$.

Step3: Substitute into Point - Slope Form

Using the point - slope form $y - y_1=m(x - x_1)$ with $x_1 = 10$, $y_1=3$ and $m = 2$, we substitute these values into the formula. We get $y - 3=2(x - 10)$.

Answer:

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