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Algebra Difficulty: Hard

$x$ $x$	$y$ $y$
$- 12$ $negative 12$	$- 45$ $negative 45$
$6$ $6$	$45$ $45$

The table shows two values of $x$ and their corresponding values of $y$ . The graph of the linear equation representing this relationship passes through the point $(\frac{1}{4}, a)$ . What is the value of $a$ ?

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Explanation

The correct answer is $StartFraction 65 Over 4 EndFraction$ . The linear relationship between $x$ and $y$ can be represented by the equation $y = m x + b$ , where $m$ and $b$ are constants. It's given in the table that when $x = - 12$ , $y = - 45$ . Substituting $negative 12$ for $x$ and $negative 45$ for $y$ in the equation $y = m x + b$ yields $- 45 = - 12 m + b$ , which can be rewritten as $- 45 + 12 m = b$ . It's also given in the table that when $x equals 6$ , $y equals 45$ . Substituting $6$ for $x$ and $45$ for $y$ in the equation $y = m x + b$ yields $45 = 6 m + b$ , which can be rewritten as $45 - 6 m = b$ . Substituting $- 45 + 12 m$ for $b$ in this equation yields $45 - 6 m = - 45 + 12 m$ . Adding $6 m$ to both sides of this equation yields $45 = - 45 + 18 m$ . Adding $45$ to both sides of this equation yields $90 equals 18 m$ . Dividing both sides of this equation by $18$ yields $5 equals m$ , or $m equals 5$ . Substituting $5$ for $m$ , $negative 12$ for $x$ , and $negative 45$ for $y$ in the equation $y = m x + b$ yields $- 45 = 5 (- 12) + b$ , or $- 45 = - 60 + b$ . Adding $60$ to both sides of this equation yields $15 equals b$ . Therefore, $m equals 5$ and $b equals 15$ . Substituting $5$ for $m$ and $15$ for $b$ in the equation $y = m x + b$ yields $y = 5 x + 15$ . Thus, the equation $y = 5 x + 15$ represents the linear relationship between $x$ and $y$ . It’s also given that the graph of the linear equation representing this relationship passes through the point $(\frac{1}{4}, a)$ . Substituting $\frac{1}{4}$ for $x$ and $a$ for $y$ in the equation $y = 5 x + 15$ yields $a = 5 (\frac{1}{4}) + 15$ , which is equivalent to $a = \frac{5}{4} + 15$ , or $a = \frac{65}{4}$ . Note that 65/4 and 16.25 are examples of ways to enter a correct answer.