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Algebra / Systems of two linear equations in two variables Difficulty: Medium

$y = \frac{2}{7} x + 3$

One of the two equations in a system of linear equations is given. The system has infinitely many solutions. If the second equation in the system is $y = m x + b$ , 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 the system has infinitely many solutions. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, $y = m x + b$ , where $m$ and $b$ are constants, have infinitely many solutions if their slopes, $m$ , are the same and if their y-coordinates of the y-intercepts, $b$ , are also the same. The first equation in the given system is $y = \frac{2}{7} x + 3$ . For this equation, the slope is $\frac{2}{7}$ and the y-coordinate of the y-intercept is $3$ . If the second equation is in the form $y = m x + b$ , then for the two equations to be equivalent, the values of $m$ and $b$ in the second equation must equal the corresponding values in the first equation. Therefore, the second equation must have a slope, $m$ , of $\frac{2}{7}$ , and a y-coordinate of the y-intercept, $b$ , of $3$ . Thus, the value of $b$ is $3$ .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.