Answer:

To find the solutions of the systems of linear equations, we will solve them using three different methods: graphing, substitution, and elimination.

1. x + y = -1

2x - y = 4

Graphing:

To graph the system of equations, we can rewrite them in slope-intercept form (y = mx + b).

Equation 1: x + y = -1

y = -x - 1

Equation 2: 2x - y = 4

y = 2x - 4

Plotting the graphs of these equations, we can see that they intersect at the point (-3, 2).

Substitution:

We can solve the system of equations using the substitution method.

From Equation 1, we have: x = -1 - y

Substituting this value of x into Equation 2:

2(-1 - y) - y = 4

-2 - 2y - y = 4

-3y - 2 = 4

-3y = 6

y = -2

Substituting the value of y back into Equation 1:

x + (-2) = -1

x - 2 = -1

x = 1

Therefore, the solution to the system of equations is x = 1 and y = -2.

Elimination:

We can solve the system of equations using the elimination method.

Multiplying Equation 1 by 2, we get:

2(x + y) = 2(-1)

2x + 2y = -2

Now, we can add this equation to Equation 2:

(2x + 2y) + (2x - y) = -2 + 4

4x + y = 2

From Equation 2, we have: y = 4 - 2x

Substituting this value of y into the equation 4x + y = 2:

4x + (4 - 2x) = 2

4x + 4 - 2x = 2

2x + 4 = 2

2x = -2

x = -1

Substituting the value of x back into Equation 2:

y = 4 - 2(-1)

y = 4 + 2

y = 6

Therefore, the solution to the system of equations is x = -1 and y = 6.

2. 2x + 2y = 6

-2x + y = 4

Graphing:

To graph the system of equations, we can rewrite them in slope-intercept form (y = mx + b).

Equation 1: 2x + 2y = 6

y = -x + 3

Equation 2: -2x + y = 4

y = 2x + 4

Plotting the graphs of these equations, we can see that they are parallel lines and do not intersect. Therefore, there is no solution to this system of equations.

Substitution:

We can solve the system of equations using the substitution method.

From Equation 2, we have: y = 2x + 4

Substituting this value of y into Equation 1:

2x + 2(2x + 4) = 6

2x + 4x + 8 = 6

6x + 8 = 6

6x = -2

x = -1/3

Substituting the value of x back into Equation 2:

y = 2(-1/3) + 4

y = -2/3 + 4

y = 10/3

Therefore, the solution to the system of equations is x = -1/3 and y = 10/3.

Elimination:

We can solve the system of equations using the elimination method.

Multiplying Equation 1 by -1, we get:

-2x - 2y = -6

Now, we can add this equation to Equation 2:

(-2x - 2y) + (-2x + y) = -6 + 4

-4x - y = -2

From Equation 2, we have: y = 4 + 2x

Substituting this value of y into the equation -4x - y = -2:

-4x - (4 + 2x) = -2

-4x - 4 - 2x = -2

-6x - 4 = -2

-6x = 2

x = -1/3

Substituting the value of x back into Equation 2:

y = 4 + 2(-1/3)

y = 4 - 2/3

y = 10/3

Therefore, the solution to the system of equations