Given a point [tex]\((-6, 10)\)[/tex] and the task to find a point on the [tex]\(x\)[/tex]-axis that lines on a line parallel to the line through this point, we can follow a step-by-step approach to solve the problem.
First, let's examine the given point and the properties of the lines in question:
1. Identifying the slope:
- A line parallel to the x-axis has a slope of 0. Therefore, the line passing through [tex]\((-6, 10)\)[/tex] that is parallel to the x-axis will also have a slope of 0. This implies such a line is a horizontal line.
2. Equation of a horizontal line:
- The equation of a horizontal line passing through any point [tex]\((x_1, y_1)\)[/tex] is [tex]\(y = y_1\)[/tex].
- Given the point [tex]\((-6, 10)\)[/tex], the equation of the line parallel to the x-axis passing through this point is [tex]\(y = 10\)[/tex].
3. Finding the point on the x-axis:
- By definition, any point on the x-axis has a [tex]\(y\)[/tex]-coordinate of 0. We need to find the x-coordinate for a point [tex]\((x, 0)\)[/tex] which lies on the x-axis.
4. Verifying the given points:
- Examining the given points:
- [tex]\((6, 0)\)[/tex]: This point lies on the x-axis.
- [tex]\((0, 6)\)[/tex]: This point does not lie on the x-axis (it lies on the y-axis).
- [tex]\((-5, 0)\)[/tex]: This point lies on the x-axis.
- [tex]\((0, -5)\)[/tex]: This point does not lie on the x-axis (it lies on the y-axis).
5. Choosing the correct points:
- The points that satisfy the condition of lying on the x-axis are [tex]\((6, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex].
Since we are interested in an ordered pair on the x-axis and both [tex]\((6, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex] meet the criteria, any can be a correct choice. According to the problem, we usually pick the first matching choice provided that meets the condition. Here, the first matching choice is [tex]\((6, 0)\)[/tex].
Thus, the ordered pair for the point on the x-axis that lies on the line parallel to the line through [tex]\((-6, 10)\)[/tex] is:
[tex]\[
\boxed{(6, 0)}
\][/tex]
