To determine the value of [tex]\( m \)[/tex] such that the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have no intersection ([tex]\( A \cap B = \varnothing \)[/tex]), let's analyze the given equations representing the two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
1. The set [tex]\( A \)[/tex] is given by the solutions to the equation [tex]\( y = 2x + 5 \)[/tex]. This represents a line with a slope of 2 and a y-intercept of 5.
2. The set [tex]\( B \)[/tex] is given by the points on the line [tex]\( y = mx \)[/tex]. This represents a line that passes through the origin with a slope of [tex]\( m \)[/tex].
To find if the two lines intersect, we need to see if there is a common point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Let's set the equations equal to each other to find points of intersection:
[tex]\[
2x + 5 = mx
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
2x + 5 = mx \implies 5 = mx - 2x \implies 5 = (m - 2)x
\][/tex]
To solve this equation, we have two situations to consider:
- If [tex]\( m \neq 2 \)[/tex], we can solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5}{m - 2}
\][/tex]
This means there is a specific [tex]\( x \)[/tex]-value at which the two lines intersect.
- If [tex]\( m = 2 \)[/tex], the equation becomes:
[tex]\[
5 = (2 - 2)x \implies 5 = 0
\][/tex]
This statement is impossible, which means there is no solution for [tex]\( x \)[/tex]. Therefore, the lines [tex]\( y = 2x + 5 \)[/tex] and [tex]\( y = 2x \)[/tex] do not intersect at any point.
Thus, the value of [tex]\( m \)[/tex] for which the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] have no intersection (i.e., [tex]\( A \cap B = \varnothing \)[/tex]) is:
[tex]\[
m = 2
\][/tex]
