Two integers, x and y, are positioned on a number line. The distance between x and 1 is tw… — ACT

Two integers, $x$ and $y$ , are positioned on a number line. The distance between $x$ and $1$ is twice the distance between $y$ and $2$ . If the distance between $x$ and $y$ is exactly $5$ units, what is the sum of all possible values of $x + y$ ?

Answer: 17

Answer

The sum of all possible values of

x + y

17

Representing the distances algebraically yields the system

|x - 1| = 2|y - 2|

and

|x - y| = 5

. Since

x

and

y

must be integers, splitting these equations into positive and negative cases yields exactly three valid integer coordinate pairs:

(13, 8)

(5, 0)

, and

(-7, -2)

. The sums (

x + y

) for these pairs are

21

5

, and

-9

, respectively. Adding these possible sums together gives a final total of

17

Step-by-Step Solution

Set up the absolute value expressions representing the distances.

|x - 1| = 2|y - 2|

and

|x - y| = 5

Distance on a number line between two points

a

and

b

is mathematically defined as

|a - b|

Split the distance condition

|x - y| = 5

into two coordinate cases.

x = y + 5

x = y - 5

An absolute value equation of the form

|A| = B

splits into

A = B

A = -B

Substitute

x = y + 5

into the first equation and solve for integer values of

y

y = 8

(which gives

x = 13

) and

y = 0

(which gives

x = 5

)

This generates the first set of valid integer coordinates satisfying all constraints.

Substitute

x = y - 5

into the first equation and solve for integer values of

y

y = -2

(which gives

x = -7

); the second algebraic option

y = 10/3

is discarded because it is not an integer

This generates the remaining valid integer coordinates satisfying all constraints.

Sum the value of

x + y

for all three valid coordinate pairs.

(13 + 8) + (5 + 0) + (-7 - 2) = 21 + 5 - 9 = 17

The question asks for the sum of all possible values of the expression

x + y

Key Concept

Using absolute value to represent distances on a number line and solving systems of absolute value equations under integer constraints.

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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