In the system of equations below, k is a positive integer that is a multiple of 9. \begin{… — SAT

In the system of equations below,

k

is a positive integer that is a multiple of 9.

\begin{aligned} 3x + 8y &= 213 \\ x - 3y &= k \end{aligned}

If the system has a solution

(x, y)

such that

x

and

y

are both positive integers, what is the value of

k

Answer: 54

Answer

Substituting

x = 3y + k

into the first equation yields

17y + 3k = 213

. Solving for

k

gives

k = 71 - \frac{17}{3}y

. For

k

to be a positive integer,

y

must be a positive multiple of 3 less than 12.5. Testing the possible values for

y

(3, 6, 9, 12) yields the possible values of

k

as 54, 37, 20, and 3. Since

k

must be a multiple of 9, the correct value is 54.

Step-by-Step Solution

Express

x

in terms of

y

and

k

using the second equation.

x = 3y + k

Isolating

x

allows for easy substitution into the first equation to eliminate one of the variables.

Substitute the expression for

x

into the first equation.

17y + 3k = 213

Substituting

x = 3y + k

into

3x + 8y = 213

yields

3(3y + k) + 8y = 213

, which simplifies to

9y + 3k + 8y = 213

and then to

17y + 3k = 213

Solve for

k

in terms of

y

and analyze the divisibility constraint.

k = 71 - \frac{17}{3}y

, where

y

is a multiple of 3

For

k

to be an integer, the fractional term

\frac{17}{3}y

must resolve to an integer, requiring

y

to be divisible by 3.

Apply the constraint that

k

must be a positive integer (

k > 0

y < 12.5

, meaning

y

can be 3, 6, 9, or 12

Setting

71 - \frac{17}{3}y > 0

yields

17y < 213

, or

y < 12.53

. The positive integer multiples of 3 in this range are 3, 6, 9, and 12.

Find the corresponding values of

k

and select the one that is a multiple of 9.

The possible values for

k

are 54, 37, 20, and 3. The only multiple of 9 is 54.

Testing the possible values of

y

gives:

y=3 \rightarrow k=54

y=6 \rightarrow k=37

y=9 \rightarrow k=20

, and

y=12 \rightarrow k=3

. Among these, 54 is the only multiple of 9.

Key Concept

Systems of Linear Equations with Integer Constraints

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic