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

Let the smaller number be denoted as \( x \) and the larger number be denoted as \( y \).

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

1. The number is equal to twice a smaller number plus 3: \[ y = 2x + 3 \]

2. The same number is equal to twice the sum of the smaller number and 1: \[ y = 2(x + 1) = 2x + 2 \]

Now we have two equations for \( y \):

1. \( y = 2x + 3 \)
2. \( y = 2x + 2 \)

Next, we can set these two equations equal to each other to find out if there is a solution: \[ 2x + 3 = 2x + 2 \]

Subtracting \( 2x \) from both sides gives: \[ 3 = 2 \]

This statement is always false, indicating that the original two equations describe parallel lines. Since parallel lines have the same slope (which is 2 in both equations) but different y-intercepts (3 and 2), they will never intersect.

Thus, there are no solutions that satisfy both equations simultaneously.

The correct conclusion is: No solutions exist because the situation describes two lines that have the same slope and different y-intercepts.