A linear system is a set of two or more linear equations with the same variables.
Example:

{x+y=10

{y=2x​

Solving the system means finding the values of the variables that make all equations true at the same time.

What does the substitution method mean?

The substitution method works by:

1. Solving one equation for one variable

2. Substituting (plugging) that expression into the other equation

3. Solving the resulting equation

4. Substituting back to find the other variable

5. Writing the solution as an ordered pair (x, y)

It’s especially useful when one equation is already solved for a variable (like $y=3x+1$).

Step-by-step example

Example 1

{y=2x+1

{x+y=7​

Step 1: Identify the equation to substitute

The first equation is already solved for $y$:

y=2x+1

Step 2: Substitute into the other equation

Replace $y$ in the second equation:

x+(2x+1)=7

Step 3: Solve for x

3x+1=7

3x=6

x=2

Step 4: Substitute back to find $y$

y=2(2)+1=5

Step 5: Write the solution

(2, 5)

✅ That pair makes both equations true.

Possible outcomes when using substitution

1. One solution → lines intersect once

2. No solution → parallel lines

3. Infinitely many solutions → same line