To find the value of [tex]\( m \)[/tex] such that the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], defined as:
- [tex]\( A = \{ \text{solutions to the equation } y = 2x + 5 \} \)[/tex]
- [tex]\( B = \{ \text{points on the line } y = mx \} \)[/tex]
do not intersect (which means [tex]\( A \cap B = \varnothing \)[/tex]), we need to determine when the lines represented by these equations are parallel and non-overlapping.
### Step-by-Step Solution:
1. Equation of Line [tex]\( A \)[/tex]:
The equation described by set [tex]\( A \)[/tex] is:
[tex]\[
y = 2x + 5
\][/tex]
2. Equation of Line [tex]\( B \)[/tex]:
The equation described by set [tex]\( B \)[/tex] is:
[tex]\[
y = mx
\][/tex]
3. Condition for Parallel Lines:
For two lines to be parallel, their slopes must be equal. The slope of line [tex]\( A \)[/tex] is 2 (from [tex]\( y = 2x + 5 \)[/tex]).
Therefore, for the lines to be parallel:
[tex]\[
m = 2
\][/tex]
4. Intersection Check:
We now need to check if the lines [tex]\( y = 2x + 5 \)[/tex] and [tex]\( y = 2x \)[/tex], with the same slope, intersect at any point.
Set the equations equal to see if there's a common point:
[tex]\[
2x + 5 = 2x
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
2x + 5 - 2x = 2x - 2x
\][/tex]
[tex]\[
5 = 0
\][/tex]
This equation is a contradiction, indicating there are no solutions where these lines intersect.
Given that the two lines with the slope [tex]\( m = 2 \)[/tex] are parallel and have no points of intersection, the solution [tex]\( A \cap B = \varnothing \)[/tex] is valid for:
[tex]\[
\boxed{2}
\][/tex]
