To find the equation of a line passing through the point [tex]\((-1,2)\)[/tex] and parallel to the line [tex]\(y = x + 4\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation given is [tex]\(y = x + 4\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Here, [tex]\(m = 1\)[/tex].
2. Use the slope of the parallel line:
Since parallel lines have the same slope, the slope of the line we need to find will also be [tex]\(1\)[/tex].
3. Apply the point-slope form of the line equation:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope. Here, [tex]\((x_1, y_1) = (-1, 2)\)[/tex] and [tex]\(m = 1\)[/tex].
4. Substitute the given point and the slope into the point-slope form:
Substituting [tex]\((-1, 2)\)[/tex] and [tex]\(m = 1\)[/tex] into the equation:
[tex]\[
y - 2 = 1(x - (-1))
\][/tex]
5. Simplify the equation:
[tex]\[
y - 2 = 1(x + 1)
\][/tex]
[tex]\[
y - 2 = x + 1
\][/tex]
Add 2 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = x + 1 + 2
\][/tex]
[tex]\[
y = x + 3
\][/tex]
Hence, the equation of the line that passes through the point [tex]\((-1, 2)\)[/tex] and is parallel to the line [tex]\(y = x + 4\)[/tex] is:
[tex]\[
y = x + 3
\][/tex]
