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Which describes the system of equations below? $\newline$ $y = x - 2$ $\newline$ $y = \frac{2}{5}x - \frac{3}{7}$ $\newline$ Choices: $\newline$ (A)consistent and independent $\newline$ (B)consistent and dependent $\newline$ (C)inconsistent

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Classify a system of equations

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Q. Which describes the system of equations below?

\newline

y = x - 2

\newline

y = \frac{2}{5}x - \frac{3}{7}

\newline

Choices:

\newline

(A)consistent and independent

\newline

(B)consistent and dependent

\newline

(C)inconsistent

Analyze Equations: To determine the type of system, we need to analyze the slopes and y-intercepts of the two equations. $\newline$ The first equation is $y = x - 2$ , which is in slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Here, the slope $m_1$ is $1$ and the y-intercept $b_1$ is $-2$ .
Identify Slopes and Intercepts: The second equation is $y = \frac{2}{5}x − \frac{3}{7}$ . It is also in slope-intercept form $y = mx + b$ . Here, the slope $m_2$ is $\frac{2}{5}$ and the y-intercept $b_2$ is $-\frac{3}{7}$ .
Determine Consistency: Since the slopes $m_1$ and $m_2$ are different ( $1 \neq \frac{2}{5}$ ), the lines are not parallel and will intersect at exactly one point. This means the system is consistent because there is at least one solution.
Check for Independence: Because the lines intersect at exactly one point, they are not the same line, and therefore the system is independent (it does not have infinitely many solutions).

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