A system of equations is a collection of two or more equations involving the same set of variables. The solution to a system is the set of values that make *all* equations true at the same time.

For a system of two linear equations in two variables ( and ), the solution corresponds to the point of intersection of the two lines when graphed.

There are three possible outcomes:

• One solution: The lines intersect at exactly one point.
• No solution: The lines are parallel and never intersect.
• Infinite solutions: The lines are the same (coincident).

Three common methods for solving systems are graphing, substitution, and elimination.

To solve by graphing, plot both equations on the same coordinate plane and identify the point where they cross.

For example, consider and :

1. Graph the line (slope , -intercept ).
2. Graph the line (slope , -intercept ).
3. The lines intersect at , so the solution is , .

Limitations: Graphing is visual and intuitive, but it can be imprecise when the solution involves fractions or very large numbers.

The substitution method solves a system by expressing one variable in terms of the other, then substituting that expression into the remaining equation.

Steps:

1. Solve one equation for one variable.
2. Substitute this expression into the other equation.
3. Solve the resulting single-variable equation.
4. Back-substitute to find the other variable.

Example: Solve and :

1. Equation 1 already gives in terms of : .
2. Substitute into equation 2: .
3. Simplify: , so .
4. Back-substitute: .

The solution is .

Substitution works well when at least one equation already has a variable isolated or can be easily solved for a variable.

The elimination method (also called the addition method) combines the two equations to cancel one variable.

Steps:

1. Write both equations in standard form: .
2. Multiply one or both equations by constants so that the coefficients of one variable are opposites.
3. Add the equations to eliminate that variable.
4. Solve the resulting single-variable equation.
5. Back-substitute to find the other variable.

Example: Solve and :

1. Notice the coefficients are and .
2. Add the equations: .
3. Simplify: , so .
4. Substitute into : , so .

The solution is .

Elimination is especially powerful when coefficients are already aligned or can be easily matched by scaling.

Not all systems have a unique solution. Two special cases arise:

No Solution (Inconsistent System): The lines are parallel. When solving, you arrive at a false statement like .

Example: and . Using substitution: → (false). No solution.

Infinite Solutions (Dependent System): The lines are identical. When solving, you arrive at a true statement like .

Example: and . Using substitution: → → (true). Infinite solutions.

The best method depends on the system's structure:

• Graphing: Good for quick estimates or when only an approximate solution is needed.
• Substitution: Best when one variable is already isolated (e.g., ) or when solving for a variable requires minimal work.
• Elimination: Best when both equations are in standard form and coefficients can be matched. This is the most efficient method for most systems.

As you practice, you will develop an intuition for which method works best in each situation.