In Exercises 49-54 , use matrices to solve the system of...

In Exercises $49-54$ , use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution.
$$
\left\{\begin{array}{l}{2 x+6 y=16} \\ {2 x+3 y=7}\end{array}\right.
$$

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Key Concepts

Back-Substitution

Back-substitution is the process used after reducing a system of equations to row echelon form. Starting with the last equation, which often contains only one variable, the known values are substituted back into the preceding equations to systematically solve for each variable in turn.

Gaussian Elimination

Gaussian elimination is an algorithmic process used to reduce a matrix to its row echelon form. This involves performing elementary row operations—swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting rows—to simplify the system, making it easier to identify the solution set or reveal any inconsistencies.

System of Linear Equations

A system of linear equations consists of multiple equations involving the same set of variables. The objective is to find values for these variables that simultaneously satisfy all given equations, often representing the intersection point(s) of lines or planes in a coordinate system.

Matrix Representation

Matrix representation is a structured way to write a system of linear equations where the coefficients of the variables are organized into a matrix. This form, typically an augmented matrix, simplifies many operations and enables the use of systematic techniques such as Gaussian elimination to find solutions.

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