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![How many solutions does the system have?
{[y=-5x+1],[y=1-5x]:}
Choose 1 answer:
(A) Exactly one solution
(B) No solutions
(C) Infinitely many solutions](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/Algebra-1/Systems%20of%20equations/Number%20of%20solutions%20to%20systems%20of%20equations/Q-16.png)
How many solutions does the system have?Choose answer:(A) Exactly one solution(B) No solutions(C) Infinitely many solutions
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Q. How many solutions does the system have?Choose answer:(A) Exactly one solution(B) No solutions(C) Infinitely many solutions
- System Solutions Overview: When does a system of equations have exactly one solution, no solutions, or infinitely many solutions?A system of linear equations can have exactly one solution if the lines intersect at one point, no solutions if the lines are parallel and do not intersect, and infinitely many solutions if the lines are coincident (the same line).
- Equations Analysis: Let's analyze the given system of equations:\{[y=-5x+1],[y=1-5x]:\}\(\newlineWe can see that both equations are in the slope-intercept form \$y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Slope and Y-intercept Comparison: Compare the slopes and y-intercepts of the two equations.\(\newline\)The first equation has a slope of \(-5\) and a y-intercept of \(1\).\(\newline\)The second equation can be rewritten as \(y = -5x + 1\), which also has a slope of \(-5\) and a y-intercept of \(1\).
- Infinite Solutions Conclusion: Since both equations have the same slope and the same \(y\)-intercept, they represent the same line. Therefore, the system of equations has infinitely many solutions because every point on the line is a solution to both equations.
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