Question

write the standard form of an equation given two points
(-1, \frac{1}{3}), (4, -3)

m = \frac{-3 - \frac{1}{3}}{4 + (-1)} \
ightarrow \frac{\frac{-10}{3}}{5} \
ightarrow \frac{-2}{3}

Explanation:

Step1: Identify the two points

The two points are \((-1, \frac{1}{3})\) and \((4, -3)\). Let \((x_1, y_1) = (-1, \frac{1}{3})\) and \((x_2, y_2) = (4, -3)\).

Step2: Calculate the slope \(m\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Substitute the values:
\(m=\frac{-3-\frac{1}{3}}{4 - (-1)}=\frac{\frac{-9 - 1}{3}}{5}=\frac{\frac{-10}{3}}{5}=\frac{-10}{3}\times\frac{1}{5}=-\frac{2}{3}\)

Step3: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((-1,\frac{1}{3})\) and \(m = -\frac{2}{3}\):
\(y-\frac{1}{3}=-\frac{2}{3}(x + 1)\)

Step4: Convert to standard form \(Ax+By = C\)

Multiply both sides by 3 to eliminate fractions:
\(3y - 1=-2(x + 1)\)
Expand the right - hand side:
\(3y-1=-2x - 2\)
Add \(2x\) to both sides and add 1 to both sides:
\(2x+3y=-2 + 1\)
\(2x+3y=-1\)

Answer:

The standard form of the equation is \(2x + 3y=-1\)