Question

write the point - slope form of the equation of the line that goes through through: (2, -5), slope = \frac{2}{3} \
\text{a } y + 5 = \frac{2}{3}(x - 2) \
\text{b } y - 2 = \frac{2}{3}(x + 5) \
\text{c } y - 5 = \frac{2}{3}(x + 2) \
\text{d } y + 2 = \frac{2}{3}(x - 5)

Explanation:

Step1: Recall Point - Slope Formula

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 \(x_1\), \(y_1\) and \(m\)

We are given the point \((2,-5)\), so \(x_1 = 2\) and \(y_1=-5\). The slope \(m=\frac{2}{3}\).

Step3: Substitute into the Formula

Substitute \(x_1 = 2\), \(y_1=-5\) and \(m = \frac{2}{3}\) into the point - slope formula \(y - y_1=m(x - x_1)\).
We get \(y-(-5)=\frac{2}{3}(x - 2)\), which simplifies to \(y + 5=\frac{2}{3}(x - 2)\).

Answer:

A. \(y + 5=\frac{2}{3}(x - 2)\)