A number is equal to twice a smaller number plus 3. The same number is equal to twice the sum of the smaller number and 1. How many solutions are possible for this situation?

Question

Infinitely many solutions exist because the two situations describe the same line.
Exactly one solution exists because the situation describes two lines that have different slopes and different y-intercepts.
No solutions exist because the situation describes two lines that have the same slope and different y-intercepts.
Exactly one solution exists because the situation describes two lines with different slopes and the same y-intercept.

Answer 1

To analyze the situation described, let's denote the larger number as \( x \) and the smaller number as \( y \).

According to the problem, we have two equations based on the descriptions:

1. The larger number \( x \) is equal to twice the smaller number plus 3: \[ x = 2y + 3 \tag{1} \]

2. The same larger number \( x \) is also equal to twice the sum of the smaller number and 1: \[ x = 2(y + 1) \tag{2} \] Simplifying this gives: \[ x = 2y + 2 \tag{2'} \]

Now, we will set the two expressions for \( x \) equal to each other: \[ 2y + 3 = 2y + 2 \]

Next, subtract \( 2y \) from both sides: \[ 3 = 2 \]

This statement is clearly false, indicating that there are no values of \( y \) that satisfy both equations simultaneously.

Therefore, since the two equations represent two lines which are parallel (they have the same slope of 2) but different y-intercepts (3 and 2, respectively), the conclusion is:

No solutions exist because the situation describes two lines that have the same slope and different y-intercepts.