We can solve a \(2\times 2\) linear system by graphing. The solution is the intersection point of the two graphs.
A linear system may be inconsistent (has no solution), dependent (has infinitely many solutions), or consistent and independent (has one solution).
Inconsistent and Dependent Systems

If an equation of the form
\begin{equation*}
0x + 0y = k \hphantom{blank}(k\ne 0)
\end{equation*}
is obtained as a linear combination of the equations in a system, the system is inconsistent.

If an equation of the form
\begin{equation*}
0x + 0y = 0
\end{equation*}
is obtained as a linear combination of the equations in a system, the system is dependent.
We can use a system of equations to solve problems involving two unknown quantities.
In economics, the price at which the supply and demand are equal is called the equilibrium price.
The solution to a \(3\times 3\) linear system is an ordered triple.
A \(3\times 3\) system in triangular form can be solved by backsubstitution.
Gaussian reduction is a generalized form of the elimination method that can be used to reduce a \(3\times 3\) linear system to triangular form.
Steps for Solving a \(3\times 3\) Linear System
Clear each equation of fractions and put it in standard form.
Choose two of the equations and eliminate one of the variables by forming a linear combination.
Choose a different pair of equations and eliminate the same variable.
Form a \(2\times 2\) system with the equations found in steps (2) and (3). Eliminate one of the variables from this \(2\times 2\) system by using a linear combination.
Form a triangular system by choosing among the previous equations. Use backsubstitution to solve the triangular system.
\(3\times 3\) linear systems may be inconsistent or dependent.
We can use a matrix to represent a system of linear equations. Each row of the matrix consists of the coefficients in one of the equations of the system.

We operate on a matrix by using the elementary row operations.
Elementary Row Operations
Multiply the entries of any row by a nonzero real number.
Add a constant multiple of one row to another row.
Interchange two rows.

We can solve a linear system by matrix reduction.
Solving a Linear System by Matrix Reduction
Write the augmented matrix for the system.
Using elementary row operations, transform the matrix into an equivalent one in upper triangular form.
Use backsubstitution to find the solution to the system.
Strategy for Matrix Reduction
If the first entry in the first row is zero, interchange that row with another. Obtain zeros in the first entries of the second and third rows by adding suitable multiples of the first row to the second and third rows.
If the second entry of the second row is zero, interchange the second and third rows. Obtain a zero in the second entry of the third row by adding a suitable multiple of the second row to the third row.
To reduce larger matrices, we start with the first row and work our way along the diagonal, using row operations to obtain nonzero entries on the diagonal and zeros below the diagonal entry.
The solutions of a linear inequality in two variables consist of a halfplane on one side of the line. The line itself is not included if the inequality is strict.
Once we have graphed the boundary line, we can decide which halfplane to shade by using a test point.
To Graph an Inequality Using a Test Point:
Graph the corresponding equation to obtain the boundary line.
Choose a test point that does not lie on the boundary line.

Substitute the coordinates of the test point into the inequality.
If the resulting statement is true, shade the halfplane that includes the test point.
If the resulting statement is false, shade the halfplane that does not include the test point.
If the inequality is strict, make the boundary line a dashed line.
The solutions to a system of inequalities include all points that are solutions to each inequality in the system. The graph of the system is the intersection of the shaded regions for each inequality in the system
To describe the solutions of a system of inequalities, it is useful to locate the vertices, or corner points, of the boundary.
Linear programming is a technique for finding the maximum or minimum value of an objective function, subject to a system of constraints.
The optimum solution occurs at one of the vertices of the set of feasible solutions.
To Solve a Linear Programming Problem:
Represent the unknown quantities by variables. Write the objective function and the constraints in terms of the variables.
Graph the solutions to the constraint inequalities.
Find the coordinates of each vertex of the solution set.
Evaluate the objective function at each vertex.
The maximum and minimum values of the objective function occur at vertices of the set of feasible solutions.
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