Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations that share the same variables. To solve a system means to find the specific values for the variables (usually $x$ and $y$) that make both equations true at the same time. Graphically, this is the point where the two lines intersect on a coordinate plane.

There are two main algebraic methods for solving systems: substitution and elimination.

Method 1: Substitution

The substitution method involves solving one equation for one variable and plugging that expression into the other equation.

Example: Solve the system: $2x + 3y = 12$ $x - y = 1$

1. Solve one equation for a variable: It is usually easiest to isolate a variable with a coefficient of $1$. Let's solve the second equation for $x$. $x = y + 1$
2. Substitute into the other equation: Replace $x$ in the first equation with $(y + 1)$. $2(y + 1) + 3y = 12$
3. Solve for the remaining variable: $2y + 2 + 3y = 12$ $5y + 2 = 12$ $5y = 10 \implies y = 2$
4. Find the second variable: Plug $y = 2$ back into your isolated equation ($x = y + 1$). $x = 2 + 1 = 3$

Solution: $(3, 2)$

Method 2: Elimination

The elimination method involves adding or subtracting the equations to cancel out one of the variables entirely.

Let's solve the exact same system using elimination: $2x + 3y = 12$ $x - y = 1$

1. Align the variables: Multiply the second equation by $3$ so the $y$ terms become opposites ($3y$ and $-3y$). $3(x - y) = 3(1) \implies 3x - 3y = 3$
2. Add the equations: Add the modified second equation to the first equation. $(2x + 3x) + (3y - 3y) = 12 + 3$ $5x = 15$
3. Solve for the variable: $x = 3$
4. Find the second variable: Substitute $x = 3$ into either of the original equations (for example, $x - y = 1$). $3 - y = 1 \implies y = 2$

Solution: $(3, 2)$

Number of Solutions

Not all systems have exactly one unique solution. There are three possibilities when solving a system of two linear equations:

1. One Solution: The lines intersect at a single point (they have different slopes).
2. No Solution: The lines are parallel and never intersect (they have the same slope, but different $y$-intercepts). Algebraically, you will end up with a false statement like $0 = 5$.
3. Infinitely Many Solutions: The equations represent the exact same line.

Example: Determine the number of solutions for: $3x + 6y = 12$ $x + 2y = 4$

Notice that if you divide every term in the first equation by $3$, you get $x + 2y = 4$. This is identical to the second equation! Because they graph as the exact same line, every point on that line is a valid solution. Therefore, this system has infinitely many solutions.