Let's analyze the statements provided about the horizontal line passing through the point [tex]\((-7,10)\)[/tex]:
1. The slope of the line is zero.
- A horizontal line has a constant y-value, meaning it does not rise or fall as it moves along the x-axis. As such, the change in y is zero for any change in x. Thus, the slope [tex]\( m \)[/tex] of a horizontal line is [tex]\( \frac{\Delta y}{\Delta x} = 0 \)[/tex].
- This statement is true.
2. Another point on the line is [tex]\((3,10)\)[/tex].
- Since the line is horizontal and passes through the point [tex]\((-7, 10)\)[/tex], all points on the line must have the same y-coordinate of 10. Therefore, [tex]\((3, 10)\)[/tex] must also lie on the line.
- This statement is true.
3. The [tex]\(y\)[/tex]-intercept of the line is -7.
- The y-intercept is the point where the line crosses the y-axis, which is the point where [tex]\(x = 0\)[/tex]. For a horizontal line, all points have the same y-coordinate. Since the given point on the line is [tex]\((-7, 10)\)[/tex], the y-coordinate is consistently 10, not -7. So, the y-intercept value should be 10, not -7.
- This statement is false.
4. The equation of the line is [tex]\(y = 10\)[/tex].
- For a horizontal line, the equation is in the form [tex]\(y = c\)[/tex], where [tex]\(c\)[/tex] is the constant y-value for all points on the line. Given that the line passes through [tex]\((-7, 10)\)[/tex], it means that every point on the line has a y-coordinate of 10. Therefore, the equation of the line is [tex]\(y = 10\)[/tex].
- This statement is true.
So, the true statements about the horizontal line passing through the point [tex]\((-7,10)\)[/tex] are:
- The slope of the line is zero.
- Another point on the line is [tex]\((3,10)\)[/tex].
- The equation of the line is [tex]\(y = 10\)[/tex].
