I don’t know how to do these..
8.
Solve the system by elimination.
{-2x + 2y + 3z = 0
{-2x – y + z = -3
{2x + 3y + 3z = 5
9. Solve the system by substitution.
{-x – y – z = -8
{-4 + 4y + 5z = 7
{2x + 2z = 4
add 1st and 3rd eqns to eliminate x
... 5y + 6z = 5 ... (A)
add 2nd and 3rd eqns to eliminate x
... 2y + 4z = 2 ... (B)
multiply B by -3/2 and add to A to elim z
... 2y = 2
solve for y, then substitute back to find x and z
solve 3rd eqn for x ... x = 2 - z
substitute for x in the 1st and 2nd eqns
... y = 6 ... (A)
... 4y + 9z = 15 ... (B)
substitute the value of y from A into B
... 4(6) + 9z = 15
solve for z, then substitute back to find x
To solve the system of equations by elimination, follow these steps:
Step 1: Multiply one or both of the equations by a constant(s) to make the coefficients of either x, y, or z the same or opposite for both equations.
In this case, you can multiply the second equation by -2 to make the coefficient of x in both equations the same.
{-2x + 2y + 3z = 0
{4x + 2y - 2z = 6
{2x + 3y + 3z = 5
Step 2: Add or subtract the two equations vertically to eliminate one variable.
Subtract the first equation from the second equation:
{2x + 2y + 3z = 0
{2x - 2y + 2z = 6
{2x + 3y + 3z = 5
{-4y + 5z = 6
Step 3: Repeat steps 1 and 2 to eliminate another variable.
Multiply the first equation by 2 and the third equation by 3 to make the coefficient of y in both equations the same:
{4x + 4y + 6z = 0
{6x + 9y + 9z = 15
Subtract the first equation from the third equation:
{6x + 9y + 9z = 15
{4x + 4y + 6z = 0
{5x + 3z = 15
Step 4: Now you have a system of two equations with two variables. Solve this system using your preferred method (substitution, elimination, or matrices).
{-4y + 5z = 6
{5x + 3z = 15
To solve the system by substitution, follow these steps:
Step 1: Solve one equation for one variable in terms of the other variable.
In this case, solve the third equation for x in terms of z:
2x + 2z = 4
2x = 4 - 2z
x = 2 - z
Step 2: Substitute the expression for the variable found in step 1 into the other equations.
Substitute x = 2 - z into the first and second equations:
-x - y - z = -8
-4 + 4y + 5z = 7
-(2 - z) - y - z = -8
-4 + 4y + 5z = 7
-2 + 2z - y - z = -8
-4 + 4y + 5z = 7
-3 + z - y = -8
-4 + 4y + 5z = 7
Step 3: Solve the resulting two equations with two variables using your preferred method (substitution, elimination, or matrices).