Linear System

SAT Questions that focus on Linear System (or System of Linear Equations) require knowledge of the following topics.

What is a Linear System?

A linear system is a set of two or more linear equations involving the same set of variables.

A linear system can be represented in a general form:

$\begin{cases}
2x+3y+z=1 \\
3x-y+z=2 \\
x+y+2z=0
\end{cases}$

or in a matrix form

$\begin{bmatrix}2 & 3 & 1\\3 & -1 & 1\\1 & 1 & 2\end{bmatrix}$.$\begin{bmatrix}x\\y\\z\end{bmatrix}$=$\begin{bmatrix}1\\2\\0\end{bmatrix}$

Elimination of Variables

The elimination of variables is the simplest method for solving linear systems. Consider the following system:

$\begin{cases}
x+y+2z=4 \\
x-y+z=1 \\
x-2y-2z=-3
\end{cases}$

Solving the first equation for x gives x = 4 -y -2z. Substituting this into the second and third equation yields:

$\begin{cases}
4-y-2z-y+z=1 \\
4-y-2z-2y-2z=-3
\end{cases}$

$\begin{cases}
-2y-z=-3 \\
-3y-4z=-7
\end{cases}$

Now solving the first equation for z gives z = 3 -2y. Substituting this into the second equation yields:

$-3y-4(3-2y)=-7$
$-3y-12+8y=-7$
$5y=5$
$y=1$

Substituting $y=1$ into the equation $-2y-z=-3$ yields:
$-2(1)-z=-3$
$-2-z=-3$
$z=1$

Substituting $y=1$ and $z=1$ into the equation $x+y+2z=4$ yields:
$x+1+2(1)=4$
$x+3=4$
$x=1$

Geometric Interpretation of a Linear System

Consider the following system:

$\begin{cases}
x+y=2 \\
x-y=0
\end{cases}$

Each of these linear equations determines a line on the xy-plane. This system has one single solution: the point where the two lines intersect (following figure):

If the equations determined parallel lines, they would never intersect, and the system would have no solution. For example:

$\begin{cases}
x+y=2 \\
x+y=3
\end{cases}$

These two equations determine two parallel lines (following figure). Since they will never intersect, the system has no solution.

If the two lines were identical, the system would have infinite solutions. That is the case of the following system:

$\begin{cases}
x+y=2 \\
2x+2y=4
\end{cases}$

Properties of Linear Systems

Independence

An equation of a linear system is independent if it cannot be derived from the others. The second equation in the following system

$\begin{cases}
x+y=2 \\
2x+2y=4
\end{cases}$

is not independent, because it is equal to the first equation times 2.

Consistency

A linear system is consistent if it has a solution.

Equivalence

Two linear systems are equivalent if they have the same solution set.

Solved SAT Practice Tests

Find Practice Tests in the following link:

SAT Practice Tests - Linear System and

Additional Practice Tests - Linear System