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$x$ $x$	$1$ $1$	$2$ $2$	$3$ $3$
$y$ $y$	$11$ $11$	$16$ $16$	$21$ $21$

The table shows three values of $x$ and their corresponding values of $y$ . Which equation represents the linear relationship between $x$ and $y$ ?

$y = 5 x + 6$

$y = 5 x + 11$

$y = 6 x + 5$

$y = 6 x + 11$

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Explanation

Choice A is correct. The linear relationship between $x$ and $y$ can be represented by the equation $y = m x + b$ , where $m$ is the slope of the line in the xy-plane that represents the relationship, and $b$ is the y-coordinate of the y-intercept. The slope can be computed using any two points on the line. The slope of a line between any two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , on the line can be calculated using the slope formula, $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . In the given table, each value of $x$ and its corresponding value of $y$ can be represented by a point $(x, y)$ . In the given table, when the value of $x$ is $1$ , the corresponding value of $y$ is $11$ and when the value of $x$ is $2$ , the corresponding value of $y$ is $16$ . Therefore, the points $left parenthesis 1 comma 11 right parenthesis$ and $left parenthesis 2 comma 16 right parenthesis$ are on the line. Substituting $left parenthesis 1 comma 11 right parenthesis$ and $left parenthesis 2 comma 16 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{16 - 11}{2 - 1}$ , or $m equals 5$ . Substituting $5$ for $m$ in the equation $y = m x + b$ yields $y = 5 x + b$ . Substituting the first value of $x$ in the table, $1$ , and its corresponding value of $y$ , $11$ , for $x$ and $y$ , respectively, in this equation yields $11 = 5 (1) + b$ , or $11 = b + 5$ . Subtracting $5$ from both sides of this equation yields $6 equals b$ . Substituting $6$ for $b$ in the equation $y = 5 x + b$ yields $y = 5 x + 6$ . Therefore, the equation $y = 5 x + 6$ represents the linear relationship between $x$ and $y$ .

Choice B is incorrect. For this relationship, when the value of $x$ is $1$ , the corresponding value of $y$ is $16$ , not $11$ .

Choice C is incorrect. For this relationship, when the value of $x$ is $2$ , the corresponding value of $y$ is $17$ , not $16$ .

Choice D is incorrect. For this relationship, when the value of $x$ is $1$ , the corresponding value of $y$ is $17$ , not $11$ .