Question

Help thanks Thanks Thanks 2 systems of equation for each one of them

Answer 1

Answer:

`\begin{gathered} 1\text{ solution: } \\ 2x\text{ + y = 4 . . .\lparen1\rparen, 3x + y = 6 . . .\lparen2\rparen} \\ 0\text{ solution:} \\ 3x\text{ - y + 5 = 0 . . \lparen1\rparen, 3x - y + 7 = 0 . . .\lparen2\rparen} \\ infinitely\text{ many solutions:} \\ 2y\text{ + x = 1 . . \lparen1\rparen, 6y + 3x = 3 . . .\lparen2\rparen} \end{gathered}`

Step-by-step explanation:

We need to give an example of 2 systems of linear equation where we have 0 solution, 1 solution and infinitely many solution. Then we will provide an explanation to determine each of them without graphing.

Example for 1 solution:

`\begin{gathered} 2x\text{ + y = 4 . . .\lparen1\rparen} \\ 3x\text{ + y = 6 . . .\lparen2\rparen} \end{gathered}`

To ascertain the two linear equations give one solution, we will have a value for x

Subtract equation (1) from (2):

`\begin{gathered} 3x\text{ - 2x + y - y = 6 - 4} \\ x\text{ + 0 = 2} \\ x\text{ = 2} \\ Since\text{ there is a value for x, it is one solution} \end{gathered}`

Example for 0 solution:

`\begin{gathered} 3x\text{ - y + 5 = 0 . . . \lparen1\rparen} \\ \text{3x - y + 7 = 0 . . .\lparen2\rparen} \end{gathered}`

To determine 0 solution system of linear equations, when we solve for the variable the left hand side will not be equal to the right hand

solving the above linear equation:

`\begin{gathered} 3x\text{ - y + 5 = 0 . . . \lparen1\rparen, 3x - y + 7 = 0 . . .\lparen2\rparen} \\ Rewriting:\text{ }3x\text{ + 5 = y \lparen1\rparen, 3x + 7 = y} \\ Equate\text{ bot equations: y = y} \\ 3x\text{ + 5 = 3x + 7} \\ collect\text{ like terms:} \\ 3x\text{ - 3x + 5 = 7} \\ 0\text{ + 5 = 0 + 7} \\ 5\text{ = 7} \\ left\text{ hand side is not equal to right hand side \lparen the values contradict each other\rparen} \end{gathered}`

Example of infintely many solutions:

`\begin{gathered} 2y\text{ + x = 1 . . . \lparen1\rparen} \\ 6y\text{ +3 x = 3 . . .\lparen2\rparen} \end{gathered}`

To determine that the solution is infinitely many solutions, the left hand side will be equal to right hand side.

`\begin{gathered} 2y\text{ + x = 1 . . . \lparen1\rparen} \\ 6y\text{ +3 x = 3 . . .\lparen2\rparen} \\ from\text{ equation \lparen1\rparen, x = 1 - 2y} \\ substitute\text{ for x in equation \lparen2\rparen:} \\ 6y\text{ + 3\lparen1 - 2y\rparen = 3} \\ 6y\text{ + 3 - 6y = 3} \\ 3\text{ = 3} \\ left\text{ hand side = right hand side} \end{gathered}`

Behaviour of the solution of the system of equations when graphed:

For a system of 2 linear equations with 1 solution:

The lines will intersect at a point. The point of intersection is 1 and it will be the solution of the system of equations

For a system of 2 linear equations with 0 solution:

The lines from both equations will be parallel to each other. Indicating the lines do not intersect. Hence, there is no solution

For a system of 2 linear equations with infinitely many solutions:

Both equation will give same line when graphed. So, the solution will be infinitely many