Solving by substitution and elimination

What a system of equations is

• A system (or simultaneous equations) is two or more equations that must hold at the same time.
• The solution is the set of values that makes every equation true together — not just one of them.
• For two linear equations in $x$ and $y$, each equation is a straight line, and the solution is the point where the lines cross.

Method 1 — substitution

• Best when one equation already gives a variable on its own (like $y = x + 1$).
• Rearrange one equation so a single variable is the subject.
• Substitute that expression into the other equation, giving one equation in one unknown.
• Solve it, then back-substitute to find the second variable.

Method 2 — elimination

• Best when the variables are lined up in columns (like $ax + by = c$).
• Match the size of one variable's coefficient in both equations, multiplying an equation through if needed.
• Add or subtract the equations to eliminate that variable.
• Solve the result, then back-substitute.

Choosing a method

• If a variable is already isolated ($y = \ldots$), reach for substitution.
• If both are in $ax + by = c$ form, elimination is usually tidier.
• Either method gives the same solution — pick whichever needs the least rearranging.