Let's break down the problem step by step:
We are given two statements about two numbers, a larger number [tex]\( L \)[/tex] and a smaller number [tex]\( S \)[/tex]:
1. The larger number is equal to twice the smaller number plus 3:
[tex]\[
L = 2S + 3
\][/tex]
2. The larger number is equal to twice the sum of the smaller number and 1:
[tex]\[
L = 2(S + 1)
\][/tex]
Let's express the second equation in terms of [tex]\( S \)[/tex]:
[tex]\[
L = 2(S + 1)
\][/tex]
Expanding the right-hand side, we get:
[tex]\[
L = 2S + 2
\][/tex]
Now we have two equations:
1. [tex]\( L = 2S + 3 \)[/tex]
2. [tex]\( L = 2S + 2 \)[/tex]
If we equate these expressions for [tex]\( L \)[/tex], we get:
[tex]\[
2S + 3 = 2S + 2
\][/tex]
Subtract [tex]\( 2S \)[/tex] from both sides:
[tex]\[
3 = 2
\][/tex]
This is a contradiction, as 3 does not equal 2. Therefore, these two equations describe two lines that have the same slope but different y-intercepts.
Thus, no solutions exist because the situation describes two lines that have the same slope and different y-intercepts.
