Pair of Linear Equations: Substitution and Elimination Methods

Worked example: substitution

Solve $x+y=14$ and $x-y=4$ by substitution.

Solution: From the second equation, $x = y+4$. Substituting into the first:

So $x = 5+4 = 9$. Checking: $9+5=14$ and $9-5=4$, both correct.

Worked example: substitution revealing infinitely many solutions

Solve $3x-y=3$ and $9x-3y=9$ by substitution.

Solution: From the first, $y = 3x-3$. Substituting into the second:

Every variable canceled out, leaving a statement that's always true. That's the signal that the second equation is just $3\times$ the first, the same line twice, with infinitely many solutions rather than one.

Worked example: elimination

Solve $x+y=5$ and $2x-3y=4$ by elimination.

Solution: Multiply the first equation by 2 so the $x$ terms match:

Subtracting $2x-3y=4$ from this eliminates $x$:

Substituting back: $x + \tfrac{6}{5} = 5 \;\Rightarrow\; x = \tfrac{19}{5}$.