Certainly! Let's go through the steps to find the equation of the straight line that passes through the point [tex]\((0,1)\)[/tex] and is perpendicular to the line [tex]\(y = -2x + 2\)[/tex].
1. Determine the slope of the given line:
The equation of the given line is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. For the line [tex]\(y = -2x + 2\)[/tex], the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line is:
[tex]\[
\text{slope} = -\frac{1}{\text{slope of the original line}}
\][/tex]
So, the slope of the perpendicular line is:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{-2} = \frac{1}{2}
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] and the point is [tex]\((0, 1)\)[/tex].
4. Substitute the values into the point-slope form:
Substitute [tex]\(m = \frac{1}{2}\)[/tex], [tex]\(x_1 = 0\)[/tex], and [tex]\(y_1 = 1\)[/tex] into the equation:
[tex]\[
y - 1 = \frac{1}{2}(x - 0)
\][/tex]
5. Simplify the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 1 = \frac{1}{2}x
\][/tex]
Add 1 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{1}{2}x + 1
\][/tex]
Therefore, the equation of the line passing through the point [tex]\((0,1)\)[/tex] and perpendicular to [tex]\(y = -2x + 2\)[/tex] is:
[tex]\[
y = \frac{1}{2}x + 1
\][/tex]
