Am I correct on these?
1.Does every line have both an x-intercept and a y-intercept?
Yes.
2.Does every line have an infinite number of lines that are parallel to the given line?
Yes.
Does every line have an infinite number of lines that are perpendicular to the given line?
No.
Vertical and horizontal lines do not have both intercepts.
If there are infinite parallel, there are infinite perpendicular.
1. You are correct. Every line does have both an x-intercept and a y-intercept. The x-intercept is the point at which the line intersects the x-axis, and the y-intercept is the point at which the line intersects the y-axis.
2. You are also correct. Every line does have an infinite number of lines that are parallel to it. This is because parallel lines have the same slope, and any given line can have infinitely many lines with the same slope.
However, for your third statement, you are incorrect. Every line does indeed have an infinite number of lines that are perpendicular to it. Perpendicular lines have negative reciprocals of each other's slopes, and any given line can have infinitely many lines with negative reciprocal slopes.
Let me explain why your answers are correct.
1. Does every line have both an x-intercept and a y-intercept?
Yes, every line has both an x-intercept and a y-intercept.
To understand why, we need to know what x-intercept and y-intercept are. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.
A line can be represented by an equation of the form y = mx + b, where m is the slope of the line and b is the y-intercept. If the line is not parallel to the y-axis (i.e., it has a slope that is not undefined), then it will intersect the y-axis at some point, giving it a y-intercept. Similarly, if the line is not parallel to the x-axis (i.e., it has a slope that is not zero), then it will intersect the x-axis at some point, giving it an x-intercept. Therefore, every line has both an x-intercept and a y-intercept.
2. Does every line have an infinite number of lines that are parallel to the given line?
Yes, every line has an infinite number of lines that are parallel to it.
To understand why, we need to know what it means for two lines to be parallel. Two lines are parallel if they have the same slope and never intersect.
Since every line can be represented by an equation of the form y = mx + b, where m is the slope, changing the value of b will give us infinitely many lines. These lines will have the same slope as the given line but may have different y-intercepts. Therefore, every line has an infinite number of lines that are parallel to it.
3. Does every line have an infinite number of lines that are perpendicular to the given line?
No, every line does not have an infinite number of lines that are perpendicular to it.
To understand why, we need to know what it means for two lines to be perpendicular. Two lines are perpendicular if their slopes are negative reciprocals of each other and they intersect at a right angle.
Since every line can be represented by an equation of the form y = mx + b, where m is the slope, changing the value of m will give us different slopes for different lines. Not all of these slopes will be negative reciprocals of each other. Therefore, not every line will have an infinite number of lines that are perpendicular to it.