To find the equation of a line that passes through the point [tex]\((2, 7)\)[/tex] and is parallel to the line [tex]\(2x - y = -1\)[/tex], follow these steps:
1. Identify the slope of the given line:
The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
For the equation [tex]\(2x - y = -1\)[/tex], we need to rewrite it in slope-intercept form:
[tex]\[
2x - y = -1 \implies -y = -2x - 1 \implies y = 2x + 1
\][/tex]
Therefore, the slope [tex]\(m\)[/tex] of the given line is [tex]\(2\)[/tex].
2. Use the point-slope form of the equation of a line:
The point-slope form of a line with slope [tex]\(m\)[/tex] passing through a point [tex]\((x_1, y_1)\)[/tex] is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(2\)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((2,7)\)[/tex]. Plugging in these values, we get:
[tex]\[
y - 7 = 2(x - 2)
\][/tex]
3. Simplify the equation:
Now, simplify the equation:
[tex]\[
y - 7 = 2x - 4
\][/tex]
Add [tex]\(7\)[/tex] to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = 2x - 4 + 7 \implies y = 2x + 3
\][/tex]
4. Re-arrange to standard form:
The equation [tex]\(y = 2x + 3\)[/tex] can be rearranged to the standard form [tex]\(Ax + By = C\)[/tex]. We get:
[tex]\[
y = 2x + 3 \implies 2x - y = -3
\][/tex]
Thus, the equation of the line passing through the point [tex]\((2,7)\)[/tex] and parallel to the line [tex]\(2x - y = -1\)[/tex] is:
[tex]\[
2x - y = -3
\][/tex]
This completes our solution.
