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How many solutions does the system of equations below have? $\newline$ $y = \frac{3}{2}x + 6$ $\newline$ $y = \frac{3}{2}x + \frac{7}{8}$ $\newline$ 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

Full solution

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

\newline

y = \frac{3}{2}x + 6

\newline

y = \frac{3}{2}x + \frac{7}{8}

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Analyze slopes: Analyze the slopes of both equations. $\newline$ The slope of the first equation $y = \frac{3}{2}x + 6$ is $\frac{3}{2}$ . $\newline$ The slope of the second equation $y = \frac{3}{2}x + \frac{7}{8}$ is also $\frac{3}{2}$ . $\newline$ Since both slopes are equal, the lines are either parallel or the same line.
Compare y-intercepts: Compare the y-intercepts of both equations. $\newline$ The y-intercept of the first equation is $6$ . $\newline$ The y-intercept of the second equation is $\frac{7}{8}$ . $\newline$ Since the y-intercepts are different, the lines are parallel and do not intersect.
Determine solutions: Determine the number of solutions. $\newline$ Since the lines are parallel and have different $y$ -intercepts, they will never intersect. $\newline$ Therefore, there are no solutions to the system of equations.

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