Mastering Linear Systems

A system of linear equations is simply a set of two or more equations with the same variables. When we graph these equations, the solution is the specific point $(x, y)$ where the lines cross. This point is special because it is the only coordinate that makes both equations true at the same time. If the lines are $y = x + 2$ and $y = -x + 4$, they intersect at $(1, 3)$. If you plug $x=1$ and $y=3$ into either equation, the math works out perfectly. We call this a consistent system because a solution exists.

Graphing is great for visuals, but it isn't always precise. The substitution method is an algebraic way to find the exact intersection. The goal is to 'plug' one equation into the other. This works best when one variable is already isolated (like $y = ...$ or $x = ...$). By replacing a variable with an equivalent expression, you turn a two-variable problem into a simple one-variable equation. For example, if $y = 2x$ and $x + y = 6$, you can substitute $2x$ for $y$ to get $x + 2x = 6$.

Not every system has exactly one solution. There are two 'weird' results you might encounter while solving algebraically: 1. No Solution: This happens with parallel lines. Algebraically, the variables will cancel out, leaving a false statement like $0 = 5$. Since $0$ never equals $5$, there is no point that satisfies both. 2. Infinitely Many Solutions: This happens when both equations describe the exact same line. The variables cancel out, leaving a true statement like $7 = 7$. This means every point on the line is a solution.