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Algebra / Linear functions Difficulty: Medium

$x$ $x$	$g (x)$ $g (x)$
$1$ $1$	$54$ $54$
$2$ $2$	$51$ $51$
$3$ $3$	$48$ $48$
$4$ $4$	$45$ $45$

For the linear function $g$ , the table shows four values of $x$ and their corresponding values of $g (x)$ . The function can be written as $g (x) = m x + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

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Explanation

Choice D is correct. It's given that for the linear function $g$ , the table shows four values of $x$ and their corresponding values of $g (x)$ . It's also given that the function can be written as $g (x) = m x + b$ , where $m$ and $b$ are constants. The table shows that when the value of $x$ is $1$ , the corresponding value of $g (x)$ is $54$ . Substituting $1$ for $x$ and $54$ for $g (x)$ in $g (x) = m x + b$ yields $54 = m (1) + b$ or, $54 = m + b$ . Subtracting $b$ from both sides of this equation yields $54 - b = m$ . The table also shows that when the value of $x$ is $2$ , the corresponding value of $g (x)$ is $51$ . Substituting $2$ for $x$ and $51$ for $g (x)$ in $g (x) = m x + b$ yields $51 = m (2) + b$ , or $51 = 2 m + b$ . Substituting $54 minus b$ for $m$ in this equation yields $51 = 2 (54 - b) + b$ . Applying the distributive property to the right-hand side of this equation yields $51 = 108 - 2 b + b$ , or $51 = 108 - b$ . Subtracting $108$ from both sides of this equation yields $- 57 = - b$ . Dividing both sides of this equation by $negative 1$ yields $57 equals b$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.