In the given system of equations, and are constants. The system has a solution of . Which choice is the value of ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 63 | Free SAT Prep | Aniko

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Question 63·200 Super-Hard SAT Math Questions·Algebra

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11kx - 6my = 99.5

5kx + 9my = 108.75

In the given system of equations, $k$ and $m$ are constants. The system has a solution of $(-4, y)$ . Which choice is the value of $k$ ?

When a solution gives one coordinate like $(-4,y)$ , substitute that coordinate immediately. Then look for a repeated product (here, $my$ ) that can be treated as a single variable, turning the problem into a standard two-equation, two-unknown system. Use elimination by scaling the equations so one variable cancels cleanly, and solve for the requested constant.

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Use the given solution point

Since $(-4,y)$ is a solution, replace $x$ with $-4$ in both equations.

Treat my as one variable

After substituting, you will see $m$ and $y$ only appear together as $my$ . Let $v=my$ to make a two-variable system.

Eliminate the my term

After substitution, choose multipliers so the coefficients of $v$ are opposites (for example, make them $-18$ and $+18$ ), then add the equations.

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Rewrite as a two-variable system

After substituting $x=-4$ , let $X$ represent $k$ and let $Y$ represent $my$ .

Graph both equations in terms of X and Y

Enter these lines:

-44X - 6Y = 99.5
-20X + 9Y = 108.75

Find the intersection

Click the intersection point of the two lines. The $X$ -coordinate of the intersection is the value of $k$ .

Step-by-step Explanation

Substitute the known x-value

Because $(-4,y)$ is a solution, substitute $x=-4$ into both equations. Let $v=my$ to simplify.

11k(-4)-6v=99.5 \quad\Rightarrow\quad -44k-6v=99.5

5k(-4)+9v=108.75 \quad\Rightarrow\quad -20k+9v=108.75

Eliminate v

Make the $v$ -coefficients opposites.

Multiply the first equation by $3$ and the second equation by $2$ :

\begin{aligned} -44k-6v&=99.5 \\ -132k-18v&=298.5 \end{aligned}

\begin{aligned} -20k+9v&=108.75 \\ -40k+18v&=217.5 \end{aligned}

Solve for k

Add the new equations to eliminate $v$ :

(-132k-18v)+(-40k+18v)=298.5+217.5

-172k=516

Divide both sides by $-172$ :

k=-3

So the correct choice is $-3$ .

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Step-by-step Explanation

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Step-by-step Explanation