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Which describes the system of equations below? $\newline$ $y = \frac{1}{3}x + 3$ $\newline$ $y = \frac{1}{3}x + \frac{10}{3}$ $\newline$ Choices: $\newline$ (A) inconsistent $\newline$ (B) consistent and dependent $\newline$ (C) consistent and independent

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Classify a system of equations

Full solution

Q. Which describes the system of equations below?

\newline

y = \frac{1}{3}x + 3

\newline

y = \frac{1}{3}x + \frac{10}{3}

\newline

Choices:

\newline

(A) inconsistent

\newline

(B) consistent and dependent

\newline

(C) consistent and independent

Slope Comparison: We have the system of equations: $\newline$ $y = \frac{1}{3}x + 3$ $\newline$ $y = \frac{1}{3}x + \frac{10}{3}$ $\newline$ Identify whether the slopes of both the equations are the same. $\newline$ In $y = \frac{1}{3}x + 3$ , the slope is $\frac{1}{3}$ . $\newline$ In $y = \frac{1}{3}x + \frac{10}{3}$ , the slope is $\frac{1}{3}$ . $\newline$ Yes, the slopes of both equations are the same.
Y-Intercept Comparison: We have: $\newline$ $y = \frac{1}{3}x + 3$ $\newline$ $y = \frac{1}{3}x + \frac{10}{3}$ $\newline$ Identify whether the y-intercepts of both the equations are the same. $\newline$ In $y = \frac{1}{3}x + 3$ , the y-intercept is $3$ . $\newline$ In $y = \frac{1}{3}x + \frac{10}{3}$ , the y-intercept is $\frac{10}{3}$ . $\newline$ Check if $3$ is equal to $\frac{10}{3}$ . $\newline$ $3$ is equal to $\frac{9}{3}$ , which is not equal to $\frac{10}{3}$ . $\newline$ No, the y-intercepts of both equations are not the same.
System Description: $y = \frac{1}{3}x + 3$ $\newline$ $y = \frac{1}{3}x + \frac{10}{3}$ $\newline$ Choose the option which describes the given system of equations. $\newline$ Since the slopes are the same but the y-intercepts are different, the lines are parallel and never intersect. $\newline$ The system of equations is inconsistent.

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