To find the equation of a line that passes through the point [tex]\((-1, 2)\)[/tex] and is parallel to the line [tex]\(y = x + 4\)[/tex], follow these steps:
1. Determine the slope of the given line:
The given line is [tex]\(y = x + 4\)[/tex]. This equation is in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope, and [tex]\(b\)[/tex] is the y-intercept. For the line [tex]\(y = x + 4\)[/tex], the slope [tex]\(m\)[/tex] is 1.
2. Use the same slope for the new line:
Since the new line is parallel to the given line, it will have the same slope. Therefore, the slope of the line we need to find is also [tex]\(m = 1\)[/tex].
3. Apply the point-slope form of the equation:
The point-slope form of the equation of a line is given by:
[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.
Given the point [tex]\((-1, 2)\)[/tex] and the slope [tex]\(m = 1\)[/tex], substitute these values into the formula:
[tex]\[
y - 2 = 1(x - (-1))
\][/tex]
Simplify the equation:
[tex]\[
y - 2 = 1(x + 1)
\][/tex]
[tex]\[
y - 2 = x + 1
\][/tex]
4. Solve for [tex]\(y\)[/tex] to get the slope-intercept form:
To get the equation into the slope-intercept form [tex]\(y = mx + b\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[
y = x + 1 + 2
\][/tex]
[tex]\[
y = x + 3
\][/tex]
Therefore, 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]
