Which describes the system of equations below? $\newline$ $y = -3x - 4$ $\newline$ $y = \frac{8}{3}x + \frac{5}{2}$ $\newline$ Choices: $\newline$ (A)inconsistent $\newline$ (B)consistent and independent $\newline$ (C)consistent and dependent

Full solution

Q. Which describes the system of equations below?

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y = -3x - 4

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y = \frac{8}{3}x + \frac{5}{2}

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Choices:

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(A)inconsistent

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(B)consistent and independent

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(C)consistent and dependent

Analyze System Type: Analyze the given system of equations to determine its type. $\newline$ We have two equations: $\newline$ $y = -3x - 4$ (Equation $1$ ) $\newline$ $y = \frac{8}{3}x + \frac{5}{2}$ (Equation $2$ ) $\newline$ To determine if the system is consistent and independent, consistent and dependent, or inconsistent, we need to compare the slopes and y-intercepts of the two lines represented by these equations.
Identify Slopes and Intercepts: Identify the slopes and $y$ -intercepts of the two lines. $\newline$ For Equation $1$ , the slope ( $m_1$ ) is $-3$ and the $y$ -intercept ( $b_1$ ) is $-4$ . $\newline$ For Equation $2$ , the slope ( $m_2$ ) is $\frac{8}{3}$ and the $y$ -intercept ( $b_2$ ) is $m_1$ $0$ .
Compare Slopes: Compare the slopes of the two lines. $\newline$ If the slopes are equal and the $y$ -intercepts are also equal, the system is consistent and dependent (the lines are the same). $\newline$ If the slopes are equal and the $y$ -intercepts are different, the system is inconsistent (the lines are parallel and never intersect). $\newline$ If the slopes are different, the system is consistent and independent (the lines intersect at one point).
Determine System Type: Determine the type of system based on the slopes and y-intercepts. $\newline$ Since $m_1 = -3$ and $m_2 = \frac{8}{3}$ , the slopes are not equal. $\newline$ Therefore, the system is consistent and independent because the lines will intersect at exactly one point.