Question 822059: given the equation 3x+9y=-8, write a second linear equation to create a system that:
• has exactly one solution. Explain your reasoning
• has no solution. Explain your reasoning
• has infinitely many solutions. Explain your reasoning
• interpret the meaning of the solution, if it exist, in the context of the graph of the following system if equations.
-5x+2y=10
10x-4y=20
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! ---
original, standard form: ax + by = c
3x + 9y = -8
slope = -a/b = -3/9 = -1/3
y-intercept = c/b = -8/9
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write second linear equation where the system has only one solution:
3x + 9y = -8
9x - 3y = -8
slope = -a/b = -9/-3 = 3
y-intercept = c/b = -8/-3 = 8/3
reason: slopes are negative reciprocal, so the lines are perpendicular and must have a unique solution, in this case the solution is:
x= -1.06666667
y= -0.533333333
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write second linear equation where the system has no solution:
3x + 9y = -8
3x + 9y = 0
slope = -a/b = -3/9 = -1/3
y-intercept = c/b = 0/9 = 0
reason: slopes are equal, so the lines are parallel, but they have distinct y-intercepts so the lines have no common solution because they never intersect
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write second linear equation where the system has infinitely many solutions:
3x + 9y = -8
6x + 18y = -16
slope = -a/b = -6/18 = -1/3
y-intercept = c/b = -16/18 = -8/9
reason: slopes are equal, so the lines are parallel, and their y-intercepts are equal, so the lines are coincident, hence have an infinite number of solutions
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-5x + 2y = 10
10x - 4y = 20
the lines have equal slopes (5/2), so the lines are parallel, but different y-intercepts (5) and (-5), so they do not intercept and hence have no common solution
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