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The elimination method is a great technique for solving systems of linear equations. It works by adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variable. Let's break down how this works using an example equation.
In our exercise, the goal was to eliminate one of the variables, either \(p\) or \(q\). Notice that the coefficients of \(q\) are opposites in the two equations: \(2q\) in the first and \(-2q\) in the second.
This tells us that adding the two equations will cancel out \(q\), leaving us with only \(p\) to solve for. The steps can be quick and simple with practice.
Here, the variable \(q\) was successfully eliminated by adding the equations together.

Solving equations is a fundamental skill in algebra. It involves finding the values of the variables that make the equation true.
When solving the simplified equation \(6p = 8\), we divided both sides by 6 to isolate \(p\). This resulted in \(p = \frac{4}{3}\).
Next, we substituted the value of \(p\) back into one of the original equations to find \(q\). This two-step approach is typical in problems like these.
We confirmed our solution by checking both \(p\) and \(q\) in the original equations. Verifying solutions ensures we solved correctly.

The substitution method is another efficient way to solve systems of equations. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
While it wasn’t the primary method used in our example, understanding it is beneficial. If we had solved for \(p\) or \(q\) first, we could then make a substitution into the other equation.
For instance, solving \(3p + 2q = 13\) for \(p\) gives \(p = \frac{13-2q}{3}\).
Substituting this into the second equation would yield \(3 \times \frac{13-2q}{3} - 2q = -5\), which can then be solved for \(q\).
This method can often be helpful when the coefficients are not easily eliminated by addition or subtraction.

Algebraic manipulation is the process of rearranging and simplifying equations using various algebraic rules. It forms the core of solving equations.
In our problem, algebraic manipulation allowed us to isolate variables and simplify equations. We performed operations like addition, subtraction, and division.
When we added the two equations, we combined like terms to eliminate \(q\). We then simplified the resulting equation to solve for \(p\).
These steps, while straightforward in this example, showcase critical techniques used in algebra to tackle more complex problems.
Remember, being comfortable with algebraic manipulation can make solving systems of equations much easier and faster.