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How many solutions does the system of equations below have? $\newline$ $y = -\frac{10}{7}x - 3$ $\newline$ $y = -\frac{10}{7}x + \frac{7}{6}$ $\newline$ Choices: $\newline$ (A)no solution $\newline$ (B)one solution $\newline$ (C)infinitely many solutions

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Find the number of solutions to a system of equations

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Q. How many solutions does the system of equations below have?

\newline

y = -\frac{10}{7}x - 3

\newline

y = -\frac{10}{7}x + \frac{7}{6}

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Analyze System of Equations: Analyze the given system of equations to determine the number of solutions. $\newline$ The system of equations is: $\newline$ $y = \frac{-10}{7}x - 3$ $\newline$ $y = \frac{-10}{7}x + \frac{7}{6}$ $\newline$ We notice that both equations have the same slope, which is $\frac{-10}{7}$ . This means that the lines are parallel unless they are the same line. To determine if they are the same line, we need to compare the y-intercepts.
Compare Y-Intercepts: Compare the $y$ -intercepts of the two equations. $\newline$ The $y$ -intercept of the first equation is $–3$ , and the $y$ -intercept of the second equation is $\frac{7}{6}$ . Since the $y$ -intercepts are different, the lines do not intersect and are not the same line.
Conclude Number of Solutions: Conclude the number of solutions based on the comparison of slopes and $y$ -intercepts. $\newline$ Since the lines are parallel and have different $y$ -intercepts, they will never intersect. Therefore, there are no solutions to the system of equations.

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