QUESTIONS (vvv)

(1) Solve the system by subsitution
2x + y = -11
3 - 4y = 11
(2) Solve the system by elimination
2x + 6 = -12
5x - 5y = 10
(3) What is the solution of the following system?
-3x - 2y = -12
9x + 6y = -9
(4) What is the solution of the following system?
x - y = 11
-x + y = -1

ANSWERS (vv)
(1)
(3, 5)
(–5, –3)
(–3, –5)
(5, 3)

(2)
(2, 1)
(0, –2)
(–2, 0)
(1, 2)

(3)
(2, 1)
no solutions
(–2, –1)
infinitely many solutions

(4)
(–3, –4)
no solutions
infinitely many solutions
(3, 4)

assuming that without typos, you meant

2x + y = -11
3x - 4y = 11

from the first, y = -2x-11
substitute that into the second to get

3x - 4(-2x-11) = 11
3x + 8x + 44 = 11
11x = -33
x = -3
so, y=-2(-3)-11 = -5

On the others, if all else fails, try checking each of the proposed solutions. If you get stuck, show us where, and we can help you along.

To solve a system of equations by substitution, follow these steps:

(1) Solve one equation for one variable in terms of the other variable.
(2) Substitute the expression for that variable in the other equation.
(3) Solve the resulting equation for the remaining variable.
(4) Substitute the value of the remaining variable back into one of the original equations to solve for the first variable.
(5) Write the solution as an ordered pair with the values of both variables.

For example, let's solve the first system of equations:

2x + y = -11 (Equation 1)
3 - 4y = 11 (Equation 2)

(1) Solve Equation 2 for y:
y = (3 - 11) / (-4) = -14 / -4 = 7/2

(2) Substitute the value of y in Equation 1:
2x + (7/2) = -11

(3) Solve for x:
2x = -11 - (7/2) = -22/2 - 7/2 = -29/2
x = (-29/2) / 2 = -29/4 = -7.25

(4) Substitute the value of x back into Equation 1:
2(-7.25) + y = -11
-14.5 + y = -11
y = -11 + 14.5 = 3.5

(5) The solution is the ordered pair (x, y): (-7.25, 3.5)

To solve a system of equations by elimination, follow these steps:

(1) Multiply one or both equations by constants so that the coefficients of one variable in both equations will be the same.
(2) Add or subtract the modified equations to eliminate one variable.
(3) Solve the resulting equation for the remaining variable.
(4) Substitute the value of the remaining variable back into one of the original equations to solve for the first variable.
(5) Write the solution as an ordered pair with the values of both variables.

Let's solve the second system of equations:

2x + 6 = -12 (Equation 1)
5x - 5y = 10 (Equation 2)

(1) Multiply Equation 1 by -5 to make the coefficient of x in both equations the same:
-10x - 30 = 60

(2) Add Equation 1 and modified Equation 2:
(-10x - 30) + (5x - 5y) = 60 + 10
-5x - 5y - 30 = 70

(3) Solve for y:
-5y = 70 + 30 + 5x
-5y = 100 + 5x
y = -20 - x

(4) Substitute the value of y back into Equation 1:
2x + 6 = -12
2x = -12 - 6
2x = -18
x = -9

(5) Substitute the values of x and y back into Equation 2 to check:
5(-9) - 5(-20 - (-9)) = 10
-45 + 5(-20 + 9) = 10
-45 + 5(-11) = 10
-45 - 55 = 10
-100 = 10

The system of equations has no solution.

To find the solution of a system of equations, you can use the same methods as described above.

The third and fourth systems of equations have multiple possible solutions, and the solutions are given in the answers provided.