Solving a System of Equations Using Substitution

Solve for $x$ and $y$ using substitution in the following system: $4x + 2y = 14$ and $3x - 5y = -22$.

Solving a system of linear equations involves finding the values of the variables that satisfy all equations in the system simultaneously. In this problem, we employ the substitution method, which is particularly useful when one of the equations can be easily solved for one variable. By restructuring one of the equations to express one variable in terms of the other, we can substitute this expression into the second equation. This reduces the system to a single equation with one variable, making it simpler to solve. Once this variable is found, it can be substituted back into the expression for the other variable.

Understanding substitution is crucial in solving systems of equations, as it builds foundational skills for handling more complex algebraic problems. It teaches how to manipulate equations and understand the relationship between variables in a system. This method not only applies to linear systems but also forms the groundwork for more advanced topics such as solving systems of nonlinear equations and further studies involving matrices and determinants in linear algebra. Students learning this technique should focus on precision in algebraic manipulation and logical deduction, which are vital tools across all mathematical problem-solving scenarios.