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Algebra Difficulty: Hard

$- x - w y = - 337$

$2 x - w y = 47$

In the given system of equations, $w$ is a constant. In the xy-plane, the graphs of these equations intersect at the point $left parenthesis q comma 19 right parenthesis$ , where $q$ is a constant. What is the value of $w$ ?

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Explanation

The correct answer is $11$ . It’s given that the graphs of the equations in the given system intersect at the point $left parenthesis q comma 19 right parenthesis$ , where $q$ is a constant. Therefore, the coordinates of this point must satisfy both equations. Substituting the point $left parenthesis q comma 19 right parenthesis$ into the first equation, $- x - w y = - 337$ , yields $- q - w (19) = - 337$ . Adding $19 w$ to both sides of this equation yields $- q = - 337 + 19 w$ , which is equivalent to $q = 337 - 19 w$ . Substituting the point $left parenthesis q comma 19 right parenthesis$ into the second equation yields $2 q - w (19) = 47$ . Substituting $337 minus 19 w$ in place of $q$ in the equation $2 q - w (19) = 47$ yields $2 (337 - 19 w) - 19 w = 47$ . Applying the distributive property to the left-hand side of this equation yields $674 - 38 w - 19 w = 47$ . Combining like terms on the left-hand side of this equation yields $674 - 57 w = 47$ . Subtracting $674$ from both sides of this equation yields $- 57 w = - 627$ . Dividing both sides of this equation by $negative 57$ yields $w equals 11$ .