Solve each system, using the method indicated.
3x + y = -7
x - y = -5 (Gaussian elimination)

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Write the system of equations as an augmented matrix: \[\left[ \begin{array}{cc|c} 3 & 1 & -7 \\ 1 & -1 & -5 \end{array} \right]\]

Use row operations to create a leading 1 in the first row, first column if necessary. In this case, the second row already has a 1 in the first column, so swap Row 1 and Row 2 to make calculations easier.

Eliminate the \(x\)-term in the second row by replacing Row 2 with (Row 2) - 3*(Row 1). This will create a zero in the first column of the second row.

Solve for \(y\) from the second row after elimination, then substitute back into the first row to solve for \(x\).

Write the solution as an ordered pair \((x, y)\) representing the values that satisfy both equations.

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주요 개념

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Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is essential before applying any solving method.

Gaussian Elimination Method

Gaussian elimination is a systematic procedure to solve systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This method simplifies the system step-by-step, making it easier to solve for each variable through back substitution.

Row Operations and Matrix Representation

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are used on the augmented matrix of the system to simplify it without changing the solution set. Mastery of these operations is crucial for effective Gaussian elimination.

Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)