Solve the System of Equations 3x-y=6 and -3x+y=-6

Solution:

Step 1:

Combine like terms by adding $3x$ to both sides of the second equation to isolate $y$.

$$y=-6+3x$$ $$3x-y=6$$

Step 2:

Substitute the expression for $y$ from the first equation into the second equation.

Step 2.1:

In the equation $3x-y=6$, replace $y$ with $-6+3x$.

$$3x-(-6+3x)=6$$ $$y=-6+3x$$

Step 2.2:

Proceed to simplify the equation.

Step 2.2.1:

Begin simplifying $3x-(-6+3x)$.

Step 2.2.1.1:

Break down the simplification process term by term.

Step 2.2.1.1.1:

Apply the distributive property to remove the parentheses.

$$3x+6-3x=6$$ $$y=-6+3x$$

Step 2.2.1.1.2:

Multiply $-1$ with $6$ to get $+6$.

$$3x+6-3x=6$$ $$y=-6+3x$$

Step 2.2.1.1.3:

Multiply $3$ by $-1$ to cancel out $3x$.

$$3x+6-3x=6$$ $$y=-6+3x$$

Step 2.2.1.2:

Combine like terms by canceling out $3x$ with $-3x$.

Step 2.2.1.2.1:

Subtract $3x$ from itself to get $0$.

$$0+6=6$$ $$y=-6+3x$$

Step 2.2.1.2.2:

Combine $0$ with $6$ to confirm the identity.

$$6=6$$ $$y=-6+3x$$

Step 3:

Eliminate any redundant equations that are identities.

$$y=-6+3x$$

Step 4:

The system is dependent, and the solution is the line $y=-6+3x$.

Knowledge Notes:

To solve a system of linear equations, one can use various methods such as substitution, elimination, or graphing. In this case, the substitution method is used. The steps involve:

1. Isolating one variable in one of the equations.

2. Substituting the expression for the isolated variable into the other equation.

3. Simplifying the resulting equation.

4. Identifying redundant or identity equations that do not contribute to the solution.

5. Interpreting the result to determine if the system has a unique solution, no solution, or infinitely many solutions (dependent system).

In this problem, the system of equations is dependent, meaning that both equations represent the same line, and thus, there are infinitely many solutions that lie on this line. The final equation, $y=-6+3x$, represents this line and is the solution to the system.