Solving a System by Elimination In Exercises 13-30, solve...

Solving a System by Elimination In Exercises
$13-30,$ solve the system by the method of elimination and
check any solutions algebraically.
$$
\left\{\begin{array}{r}{\frac{x-1}{2}+\frac{y+2}{3}=4} \\ {x-2 y=5}\end{array}\right.
$$

Key Concepts

Algebraic Verification (Solution Checking)

After obtaining a solution for a system of equations, it is important to verify that the solution satisfies each original equation. This step, known as algebraic verification, involves substituting the found values back into the original equations. It confirms the accuracy of the solution and ensures that no mistakes were made during the elimination or manipulation process.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find a solution that satisfies all equations simultaneously. This is a fundamental concept in algebra, as it enables the determination of variable values that make all equations true at the same time.

Method of Elimination

The elimination method involves manipulating the equations in a system in order to cancel out one of the variables. This is typically achieved by aligning the coefficients of one variable and then adding or subtracting the equations, resulting in a single equation with one variable. This technique is key for simplifying and solving systems of equations effectively.

Clearing Fractions

Clearing fractions is a preliminary step often used when equations contain fractional terms. Multiplying both sides of an equation by the least common denominator eliminates the fractions, producing an equivalent equation with integer coefficients. This simplification makes further manipulation of the equation, such as elimination or substitution, much easier.

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