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How many solutions does the system of equations below have? $\newline$ $y = 7x - 8$ $\newline$ $y = \frac{7}{2}x + \frac{1}{9}$ $\newline$ 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 = 7x - 8

\newline

y = \frac{7}{2}x + \frac{1}{9}

\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 $y = 7x − 8$ is $7$ . $\newline$ The slope of the second equation $y = \frac{7}{2}x + \frac{1}{9}$ is $\frac{7}{2}$ . $\newline$ Since the slopes are different $7 \neq \frac{7}{2}$ , the lines are not parallel.
Determine intersection: Determine if the lines intersect. $\newline$ Since the slopes are different, the lines will intersect at exactly one point. $\newline$ This means there is one unique solution to the system of equations.
Confirm y-intercepts: Confirm that the y-intercepts are different. $\newline$ The y-intercept of the first equation is $-8$ . $\newline$ The y-intercept of the second equation is $\frac{1}{9}$ . $\newline$ Since the y-intercepts are different, it confirms that the lines are not the same and they intersect at one point.

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