Solve the systems of equations in Exercises 8-11 by...

Solve the systems of equations in Exercises $8-11$ by elimination, and check your solutions. Give the following information: $\cdot$ which variable you eliminated $\cdot$ whether you added or subtracted equations $\bullet$ the solution $$ \begin{array}{l}{3 p+2 q=13} \\ {3 p-2 q=-5}\end{array} $$

Solve the systems of equations in Exercises $8-11$ by elimination, and check your solutions. Give the following information:
$\cdot$ which variable you eliminated
$\cdot$ whether you added or subtracted equations
$\bullet$ the solution
$$
\begin{array}{l}{3 p+2 q=13} \\ {3 p-2 q=-5}\end{array}
$$

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Key Concepts

System of Linear Equations

This concept involves a set of equations with multiple unknowns, where the solution is the common set of values that satisfies all the equations simultaneously. It forms the basis for modeling situations in algebra and linear algebra and is often solved by methods like substitution, elimination, or graphical approaches.

Elimination Method

The elimination method is a systematic procedure for solving systems of linear equations by adding or subtracting equations to cancel out one of the variables. This simplifies the system into a single equation with one variable, making it easier to solve the problem step-by-step before back-substituting to find the other variable.

Addition/Subtraction of Equations

This technique involves adding or subtracting entire equations in a system to remove one variable entirely. Choosing the correct operation is critical and depends on noticing coefficients that are additive inverses, which ultimately reduces the complexity of the system.

Solution Verification

After obtaining potential solutions for the variables, it is essential to substitute them back into the original equations to check if they satisfy all the equations. This verification step ensures the accuracy of the solution and confirms that no computational errors were made during the elimination process.

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