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

\newline

y = -x - \frac{3}{4}

\newline

Choices:

\newline

(A)consistent and dependent

\newline

(B)inconsistent

\newline

(C)consistent and independent

Compare Slopes: We are given the system of equations: $\newline$ $y = -x - \frac{3}{4}$ $\newline$ $y = -x - \frac{3}{4}$ $\newline$ First, we need to compare the slopes of both equations. $\newline$ In $y = -x - \frac{3}{4}$ , the slope is $-1$ . $\newline$ In $y = -x - \frac{3}{4}$ , the slope is also $-1$ .
Compare Y-Intercepts: Next, we compare the y-intercepts of both equations. $\newline$ In $y = -x - \frac{3}{4}$ , the y-intercept is $-\frac{3}{4}$ . $\newline$ In $y = -x - \frac{3}{4}$ , the y-intercept is also $-\frac{3}{4}$ .
Identical Lines: Since both the slope and $y$ -intercept of the two equations are the same, the lines represented by these equations are identical. This means that every solution to one equation is also a solution to the other, and there are infinitely many solutions.
Consistent and Dependent: Therefore, the system of equations is consistent because there are solutions, and it is dependent because the equations represent the same line, and thus all solutions are shared between the two equations.

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