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How many solutions does the system have? {[8x+2y=14],[8x+2y=4]:} Choose 1 answer: (A) Exactly one solution (B) No solutions (c) Infinitely many solutions

How many solutions does the system have?\newline{8x+2y=148x+2y=4 \left\{\begin{array}{l} 8 x+2 y=14 \\ 8 x+2 y=4 \end{array}\right. \newlineChoose 11 answer:\newline(A) Exactly one solution\newline(B) No solutions\newline(C) Infinitely many solutions

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Q. How many solutions does the system have?\newline{8x+2y=148x+2y=4 \left\{\begin{array}{l} 8 x+2 y=14 \\ 8 x+2 y=4 \end{array}\right. \newlineChoose 11 answer:\newline(A) Exactly one solution\newline(B) No solutions\newline(C) Infinitely many solutions
  1. Analyze Equations: Let's analyze the system of equations:\newline{8x+2y=148x+2y=4 \begin{cases} 8x + 2y = 14 \\ 8x + 2y = 4 \end{cases} \newlineWe can see that both equations have the same coefficients for xx and yy, but different constant terms. This suggests that the lines represented by these equations are parallel.
  2. Compare Slopes: To confirm if the lines are indeed parallel, we can compare the slopes of the lines. The slope-intercept form of a line is y=mx+by = mx + b, where mm is the slope. Let's convert the first equation to slope-intercept form by isolating yy:\newline8x+2y=14 8x + 2y = 14 \newline2y=−8x+14 2y = -8x + 14 \newliney=−4x+7 y = -4x + 7 \newlineThe slope of the first line is −4-4.
  3. Convert to Slope-Intercept Form: Now, let's convert the second equation to slope-intercept form:\newline8x+2y=4 8x + 2y = 4 \newline2y=−8x+4 2y = -8x + 4 \newliney=−4x+2 y = -4x + 2 \newlineThe slope of the second line is also −4-4.
  4. Confirm Parallel Lines: Since both lines have the same slope but different yy-intercepts (77 and 22, respectively), they are parallel and will never intersect. Therefore, the system of equations has no solutions.

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