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).

Question

Accepted Answer

∻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.⚹◊What are the conditions for a system of linear equations to have no solution, one solution, or infinitely many solutions?⍭ Generate me a similar question◊

Sia