Consider the system of linear equations below, where k is a constant: \begin{aligned} 3x -… — SAT

Consider the system of linear equations below, where $k$ is a constant:

\begin{aligned} 3x - 2y &= 7 \\ kx + 3y &= 18 \end{aligned}

If the system has a solution $(x, y)$ such that $x + y = 4$ , what is the value of $k$ ?

The correct answer is

5

. To find the value of

k

, we first express

x

in terms of

y

using the given constraint equation:

x = 4 - y

. Next, we substitute this expression into the first equation of the system:

3(4 - y) - 2y = 7

. Distributing and combining like terms yields

12 - 5y = 7

. Solving for

y

gives

y = 1

. Substituting this back into the constraint equation gives

x = 4 - 1 = 3

. Finally, we substitute the solution point

(3, 1)

into the second equation:

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

Express one variable in terms of the other using the given constraint equation

x + y = 4

x = 4 - y

This allows us to substitute the expression for

x

into the first equation to reduce it to a single variable.

Substitute

x = 4 - y

into the first equation

3x - 2y = 7

and solve for

y

3(4-y) - 2y = 7 \implies 12 - 3y - 2y = 7 \implies 12 - 5y = 7 \implies -5y = -5 \implies y = 1

By solving this linear equation, we find the unique y-coordinate of the system's solution.

Substitute the value of

y = 1

back into the constraint equation to find

x

x = 4 - 1 = 3

This determines the x-coordinate of the solution point, giving us the full solution

(3, 1)

Substitute the point

(3, 1)

into the second equation

kx + 3y = 18

to solve for the constant

k

k(3) + 3(1) = 18 \implies 3k + 3 = 18 \implies 3k = 15 \implies k = 5

Since the solution must satisfy all equations in the system, we can solve for

k

using the coordinates we found.

Solving systems of linear equations under linear constraints using algebraic substitution.

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