To determine the equation of the line passing through the points [tex]\(A(0, -4)\)[/tex] and [tex]\(B(6, 2)\)[/tex], we need to follow these steps:
1. Calculate the Slope:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
y_1 = -4, \quad y_2 = 2, \quad x_1 = 0, \quad x_2 = 6
\][/tex]
[tex]\[
m = \frac{2 - (-4)}{6 - 0} = \frac{2 + 4}{6} = \frac{6}{6} = 1
\][/tex]
2. Determine the Y-Intercept:
The slope-intercept form of a line is:
[tex]\[
y = mx + b
\][/tex]
Given that the slope [tex]\(m\)[/tex] is 1 and using the point [tex]\(A(0, -4)\)[/tex]:
[tex]\[
y = 1 \cdot x + b
\][/tex]
Since the point [tex]\(A\)[/tex] is on the line, substituting [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[
-4 = 1 \cdot 0 + b \implies b = -4
\][/tex]
Therefore, the equation of the line is:
[tex]\[
y = x - 4
\][/tex]
3. Verify the Provided Options:
Let's compare this equation with the given options:
- [tex]\( y - 4 = 3x \)[/tex]
[tex]\[
\text{Rewriting: } y = 3x + 4 \quad (\text{This has a slope of 3, not 1})
\][/tex]
- [tex]\( y - 2 = 3(x - 6) \)[/tex]
[tex]\[
\text{Expanding: } y - 2 = 3x - 18 \implies y = 3x - 16 \quad (\text{This has a slope of 3, not 1})
\][/tex]
- [tex]\( y + 4 = x \)[/tex]
[tex]\[
\text{Rewriting: } y = x - 4 \quad (\text{This matches our equation})
\][/tex]
- [tex]\( y + 6 = x - 2 \)[/tex]
[tex]\[
\text{Rewriting: } y = x - 8 \quad (\text{This does not match our equation})
\][/tex]
The equation that correctly represents the line passing through the points [tex]\(A(0, -4)\)[/tex] and [tex]\(B(6, 2)\)[/tex] is:
[tex]\[ \boxed{y + 4 = x} \][/tex]
