Use the addition-subtraction method to find all solutions of...

Use the addition-subtraction method to find all solutions of each system of equations.
$$\left\{\begin{array}{l}
\frac{1}{4} x-\frac{1}{3} y=4 \\
\frac{2}{7} x-\frac{1}{7} y=\frac{1}{10}
\end{array}\right.$$

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Key Concepts

Elimination Method

The elimination method, also known as the addition-subtraction method, is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one or more variables. This process simplifies the system into a form where individual variables can be solved more easily, reducing the problem to one or more single-variable equations.

Clearing Fractions

Clearing fractions is a strategy used to simplify equations that contain fractional coefficients. By multiplying all terms of an equation by the least common multiple of the denominators, the fractions are eliminated, resulting in an equivalent equation with integer coefficients. This transformation makes the process of elimination and other algebraic operations more straightforward.

System of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables. The solution to the system is the set of all variable values that satisfy every equation simultaneously. This concept is fundamental in algebra and is widely used to model real-world situations where multiple relationships or conditions intersect.

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