A system of equations has no solution. If [tex]y=8x+7[/tex] is one of the equations, which could be the other equation?
A. [tex]2y=16x+14[/tex]
B. [tex]y=8x-7[/tex]
C. [tex]y=-8x+7[/tex]
D. [tex]2y=-16x-14[/tex]
Answer (2)
Recognize that a system of equations has no solution when the lines are parallel with different y-intercepts.
Identify the slope and y-intercept of the given equation y = 8 x + 7 .
Check each option to see which equation has the same slope but a different y-intercept.
Conclude that y = 8 x − 7 results in a system with no solution because it has the same slope (8) but a different y-intercept (-7).
The other equation is y = 8 x − 7 .
Explanation
Understanding the Problem We are given a system of equations where one equation is y = 8 x + 7 . We need to determine which of the provided equations, when paired with the given equation, results in a system with no solution.
Condition for No Solution For a system of linear equations to have no solution, the lines must be parallel but have different y-intercepts. Parallel lines have the same slope.
Identifying Slope and Intercept The given equation is y = 8 x + 7 . This is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope m = 8 and the y-intercept b = 7 .
Analyzing the Options Now, let's examine each of the provided equations to see if they have the same slope but a different y-intercept.
2 y = 16 x + 14 . Dividing both sides by 2, we get y = 8 x + 7 . This equation is identical to the given equation, so it represents the same line. Therefore, this system has infinitely many solutions, not no solution.
y = 8 x − 7 . This equation has a slope of 8 and a y-intercept of -7. Since the slopes are the same (8) but the y-intercepts are different (7 and -7), this system has no solution.
y = − 8 x + 7 . This equation has a slope of -8 and a y-intercept of 7. Since the slopes are different (8 and -8), this system has one solution.
2 y = − 16 x − 14 . Dividing both sides by 2, we get y = − 8 x − 7 . This equation has a slope of -8 and a y-intercept of -7. Since the slopes are different (8 and -8), this system has one solution.
Determining the Correct Equation The equation y = 8 x − 7 has the same slope as y = 8 x + 7 but a different y-intercept. Therefore, the system of equations
y = 8 x + 7 y = 8 x − 7
has no solution.
Examples
Understanding systems of equations with no solution is crucial in various real-world scenarios. For instance, consider a situation where two companies offer services with linearly increasing costs. If the cost equations have the same rate of increase (slope) but different initial fees (y-intercepts), there will be no point at which the services cost the same. This concept applies in economics, engineering, and resource allocation, where identifying conflicting constraints is essential for effective decision-making. The ability to recognize parallel lines and their implications helps in avoiding scenarios where solutions are impossible.
The other equation that could lead to a system with no solution when paired with y = 8 x + 7 is y = 8 x − 7 . This option has the same slope but a different y-intercept, making the lines parallel. Therefore, the answer is option B: y = 8 x − 7 .
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