To find the equation of the line that passes through the points [tex]\( (2,5) \)[/tex] and [tex]\( (6,9) \)[/tex], we need to determine the slope and the y-intercept of the line.
### Step 1: Calculate the slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates [tex]\( (2,5) \)[/tex] and [tex]\( (6,9) \)[/tex] into the formula:
[tex]\[
m = \frac{9 - 5}{6 - 2} = \frac{4}{4} = 1
\][/tex]
### Step 2: Calculate the y-intercept
The equation of the line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
We already know that [tex]\( m = 1 \)[/tex]. To find [tex]\( b \)[/tex], we can use one of the given points. Let's use the point [tex]\( (2, 5) \)[/tex].
Substitute [tex]\( m = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( y = 5 \)[/tex] into the equation:
[tex]\[
5 = 1(2) + b
\][/tex]
[tex]\[
5 = 2 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 5 - 2 = 3
\][/tex]
### Step 3: Write the equation
Now that we have the slope [tex]\( m = 1 \)[/tex] and the y-intercept [tex]\( b = 3 \)[/tex], we can write the equation of the line:
[tex]\[
y = 1x + 3
\][/tex]
### Step 4: Match with the given options
We compare this equation with the provided choices:
- [tex]\( y = x + 3 \)[/tex]
- [tex]\( y = x - 3 \)[/tex]
- [tex]\( y = 3x + 3 \)[/tex]
- [tex]\( y = 3x - 3 \)[/tex]
The correct equation [tex]\( y = x + 3 \)[/tex] matches the first option.
Therefore, the equation that represents the line GH is:
[tex]\[
\boxed{y = x + 3}
\][/tex]
