SAT Question 38c90632: Equivalent Expressions answer and explanation with Desmos solution

The given expression $\frac{p}{x}+\frac{w}{x+51}$ can be rewritten with the common denominator $x(x+51)$. To add the two terms of this expression, $\frac{p}{x}$ can be rewritten as $\frac{p(x+51)}{x(x+51)}$ and $\frac{w}{(x+51)}$ can be rewritten as $\frac{wx}{x(x+51)}$. Therefore, the expression $\frac{p}{x}+\frac{w}{(x+51)}$ can be rewritten as $\frac{p(x+51)}{x(x+51)}+\frac{wx}{x(x+51)}$, which is equivalent to $\frac{px+51p+wx}{x(x+51)}$, or $\frac{(p+w)x+51p}{x(x+51)}$. Since $\frac{29x+102}{x(x+51)}$ is equivalent to $\frac{(p+w)x+51p}{x(x+51)}$ and these expressions have equivalent denominators, these expressions must also have equivalent numerators. It follows that $29x+102=(p+w)x+51p$. In this equation, the x-coefficients must be equal and the constant terms must be equal. Therefore, $p+w=29$ and $51p=102$. Dividing both sides of $51p=102$ by $51$ yields $p=2$. Substituting $2$ for $p$ in $p+w=29$ yields $2+w=29$. Subtracting $2$ from both sides of this equation yields $w=27$. Therefore, the value of $w$ is $27$.