Question

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?
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

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

Explanation:

Let x be the number.

Let y denote the smaller number.

It is given that "A number is equal to twice a smaller number plus 3", which can be written in expression as,

$x=2y+3$

Also, it is given that, "The same number is equal to twice the sum of the smaller number and 1", which can be written as,

$x=2(y+1)\\x=2y+2$

Thus, to determine the number of solutions for these two equations, let us solve the equation by the method of substitution.

Hence, we have,

$2y+3=2y+2$

Simplifying, we have,

$3\neq 2$

Hence, we can see from the equation that there is no solution because it has same slope and different y-intercepts.

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

Hence, Option C is the correct answer.