Question

line gh passes through points (2, 5) and (6, 9). which equation represents line gh?
○ $y = x + 3$
○ $y = x - 3$
○ $y = 3x + 3$
○ $y = 3x - 3$

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For points \((2, 5)\) and \((6, 9)\), \(x_1 = 2\), \(y_1 = 5\), \(x_2 = 6\), \(y_2 = 9\).
So \(m=\frac{9 - 5}{6 - 2}=\frac{4}{4}=1\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Let's use the point \((2, 5)\) and \(m = 1\).
Substitute into the formula: \(y - 5=1\times(x - 2)\).
Simplify the equation: \(y-5=x - 2\), then \(y=x - 2 + 5=x + 3\).
We can also check by substituting the points into the given options.
For the option \(y=x + 3\):
When \(x = 2\), \(y=2 + 3=5\) (matches the point \((2,5)\)).
When \(x = 6\), \(y=6 + 3=9\) (matches the point \((6,9)\)).
For \(y=x - 3\):
When \(x = 2\), \(y=2-3=- 1
eq5\), so this option is wrong.
For \(y = 3x+3\):
When \(x = 2\), \(y=3\times2 + 3=9
eq5\), so this option is wrong.
For \(y = 3x-3\):
When \(x = 2\), \(y=3\times2-3 = 3
eq5\), so this option is wrong.

Answer:

A. \(y=x + 3\)