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Math

Posted 6 months ago

`Question #1: Create a system of linear equations with no solutions, one solution, and infinitely many solutions. Explain how you determine the number of solutions without solving them. Then, verify the solutions by solving them, both algebraically and graphically (by free-hand).`

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To create a system with no solutions, the equations must be parallel and have different y-intercepts

step 2

Example: $x + y = 2$ and $x + y = 5$. These lines are parallel and will never intersect

step 3

To create a system with one solution, the equations must intersect at exactly one point

step 4

Example: $x + y = 2$ and $x - y = 0$. These lines intersect at one point, which is the solution

step 5

To create a system with infinitely many solutions, the equations must be the same line or multiples of each other

step 6

Example: $x + y = 2$ and $2x + 2y = 4$. These are the same line, so they have infinitely many solutions

step 7

Algebraically solving the system with no solutions: Since the equations are the same, subtracting one from the other gives $0 = 3$, which is a contradiction

step 8

Algebraically solving the system with one solution: Adding the two equations $x + y = 2$ and $x - y = 0$ gives $2x = 2$, so $x = 1$. Substituting $x = 1$ into $x - y = 0$ gives $y = 1$

step 9

Algebraically solving the system with infinitely many solutions: Dividing the second equation by 2 gives $x + y = 2$, which is the same as the first equation

step 10

Graphically solving the systems: Plot the lines on a coordinate plane. For no solutions, the lines are parallel. For one solution, the lines intersect at a point. For infinitely many solutions, the lines overlap

[question number] Answer

No solutions: System $x + y = 2$, $x + y = 5$. One solution: System $x + y = 2$, $x - y = 0$. Infinitely many solutions: System $x + y = 2$, $2x + 2y = 4$.

Key Concept

Determining the number of solutions for a system of linear equations without solving

Explanation

Systems of linear equations can have no solutions if the lines are parallel and different, one solution if they intersect at a single point, or infinitely many solutions if they represent the same line or are multiples of each other. This can be determined by comparing the slopes and y-intercepts of the equations.

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