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How many solutions does the system of equations below have? $\newline$ $y = -3x + 4$ $\newline$ $y = \frac{1}{5}x - \frac{8}{3}$ $\newline$ 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 = -3x + 4

\newline

y = \frac{1}{5}x - \frac{8}{3}

\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 $y = -3x + 4$ is $-3$ . $\newline$ The slope of the second equation $y = \frac{1}{5}x - \frac{8}{3}$ is $\frac{1}{5}$ . $\newline$ Since the slopes are different, the lines are not parallel.
Determine Intersection: Determine if the lines intersect. $\newline$ Because the slopes are different, the lines must intersect at exactly one point.
Conclude Number of Solutions: Conclude the number of solutions. Since the lines intersect at exactly one point, the system of equations has $1$ solution.

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