Use elimination to solve each system of equations. Check...

Use elimination to solve each system of equations. Check your solution.
$\left\{\begin{aligned} \frac{1}{5} x-\frac{3}{2} y &=4 \\-\frac{2}{3} x+\frac{1}{2} y &=-\frac{13}{3} \end{aligned}\right.$
(Hint: Clear fractions first to simplify the arithmetic.)

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more equations with the same variables. The goal is to find the set of variable values that simultaneously satisfy all the equations in the system. This forms the basis for many real-world applications where multiple conditions must be met concurrently.

Clearing Fractions

Clearing fractions is a method used to simplify equations by multiplying both sides by the least common multiple of the denominators. This process transforms the equation into an equivalent one without fractions, making the arithmetic more straightforward and reducing errors during further manipulations.

Elimination Method

The elimination method is a systematic approach to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. By eliminating a variable, the system reduces to a single equation with one unknown, which can then be solved directly. This method is particularly effective when the coefficients are carefully chosen or adjusted.

Solution Verification

Solution verification is the process of checking the obtained solution by substituting the values back into the original equations. This step ensures that the solution satisfies all the equations in the system, confirming its correctness and preventing potential miscalculations from earlier steps.

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