Question: What is a $2\times 2$ linear system?
Answer: A $2 \times 2$ linear system is 2 equations in 2 unknowns.
Example: $$\left\{\begin{array}{l}-3x-5y=-1\\x+2y=1\\\end{array}\right\}$$
Question: What is a solution to a $2 \times 2$ linear system?
Answer: A solution to a $2 \times 2$ linear system is an ordered pair which makes both equations true.
Example: $$\left\{\begin{array}{l}-3x-5y=-1\\x+2y=1\\\end{array}\right\}$$
The point $(-3,2)$ is a solution because when $x=-3$ and $y=2$, both equations are satisfied.
Question: How do we find solutions to linear systems?
Answer: Very carefully. There are several ways.
One way to find solutions is to graph both lines...
Example: Solve the following system by graphing both equations. $$\left\{\begin{array}{l}-3x-5y=-1\\x+2y=1\\\end{array}\right\}$$
Another way is to use a table....
Example: Solve the system $\left\{\begin{array}{l}y=-\frac{1}{2}x+1\\y=\frac{1}{2}x+2\\\end{array}\right\}$ by using the table of below:
$\begin{array}{c|c|c}x & -\frac{1}{2}x+1 & \frac{1}{2}x+2\\ \hline-5 & 3.5 & -0.5 \\-4.5 & 3.25 & -0.25 \\-4 & 3 & 0 \\-3.5 & 2.75 & 0.25 \\-3 & 2.5 & 0.5 \\-2.5 & 2.25 & 0.75 \\-2 & 2 & 1 \\-1.5 & 1.75 & 1.25 \\-1 & 1.5 & 1.5 \\-0.5 & 1.25 & 1.75 \\0 & 1 & 2 \\0.5 & 0.75 & 2.25 \\1 & 0.5 & 2.5 \\1.5 & 0.25 & 2.75 \\2 & 0 & 3 \\2.5 & -0.25 & 3.25 \\3 & -0.5 & 3.5 \\3.5 & -0.75 & 3.75 \\4 & -1 & 4 \\4.5 & -1.25 & 4.25 \\5 & -1.5 & 4.5 \\\end{array}$
Question: How many solutions can a linear system have?
Answer: That depends on the system.
Let's think about what can happen with two lines...
Vocab:
A linear system with one solution is called a consistent system of independent equations.
A linear system with no solutions is called an inconsistent system.
A linear system with infinitely many solutions is called a consistent system of dependent equations.
Example: Classify the following system according to its number of solutions. $$\left\{\begin{array}{l}y=-\frac{1}{2}x-2\\y=-\frac{1}{2}x+3\\\end{array}\right\}$$
Example: Classify the following system according to its number of solutions. $$\left\{\begin{array}{l}\frac{3}{2}x+y=4\\-3x-2y=-8\\\end{array}\right\}$$
Question: What happens if I want to solve a system graphically but you don't give me a graph? D:
Answer: Use technology! Here's an online graphing calculator which calculates intersection points.