To find the linear equation of a line that passes through the point [tex]\((-6, -1)\)[/tex] and is parallel to the line [tex]\(y = -\frac{2}{3}x + 1\)[/tex], we will follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(y = -\frac{2}{3}x + 1\)[/tex]. The coefficient of [tex]\(x\)[/tex] in this equation is the slope (denoted [tex]\(m\)[/tex]). Hence, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{2}{3}\)[/tex].
2. Determine the slope of the parallel line:
Since parallel lines have the same slope, the slope of the new line will also be [tex]\(-\frac{2}{3}\)[/tex].
3. Use the point-slope formula to formulate 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. For our line, we use the point [tex]\((-6, -1)\)[/tex] and the slope [tex]\(-\frac{2}{3}\)[/tex].
Substituting in the values, we get:
[tex]\[
y - (-1) = -\frac{2}{3}(x - (-6))
\][/tex]
Simplifying the equation:
[tex]\[
y + 1 = -\frac{2}{3}(x + 6)
\][/tex]
4. Distribute the slope on the right-hand side:
[tex]\[
y + 1 = -\frac{2}{3}x - \frac{2}{3} \times 6
\][/tex]
[tex]\[
y + 1 = -\frac{2}{3}x - 4
\][/tex]
5. Isolate [tex]\( y \)[/tex] to convert to slope-intercept form:
[tex]\[
y = -\frac{2}{3}x - 4 - 1
\][/tex]
[tex]\[
y = -\frac{2}{3}x - 5
\][/tex]
Therefore, the linear equation in slope-intercept form that passes through the point [tex]\((-6, -1)\)[/tex] and is parallel to the line [tex]\(y = -\frac{2}{3}x + 1\)[/tex] is:
[tex]\[
y = -\frac{2}{3}x - 5
\][/tex]
