Solve a system of two linear equations by multiplying to match coefficients, then adding or subtracting to eliminate a variable — with every algebraic step and a graph of the intersection.
a₁x + b₁y = c₁
a₂x + b₂y = c₂
The goal is to multiply one or both equations by constants so that a chosen variable has the same (or opposite) coefficient in both equations. Then add or subtract to cancel that variable.
1. Choose a variable — pick x or y (whichever has simpler multipliers).
2. Find the LCM of the two coefficients, then multiply each equation to reach that LCM.
3. Add or subtract the equations — the chosen variable disappears, leaving one equation in one unknown.
4. Solve the remaining one-variable equation, then substitute back to find the other variable.
5. Verify by substituting both values into each original equation.
Use elimination when coefficients are integers and one divides the other, or when both variables appear in both equations with similar magnitudes.
Elimination shines when the system has "nice" integer coefficients and no equation is already solved for one variable. Matching coefficients is faster than solving for y and substituting an ugly fraction.
Substitution is easier when one equation is already in the form x = … or y = …, or has a coefficient of 1 or −1, making isolation trivial.
A focused tutoring session makes elimination and substitution click. Work through real problems with personalized, step-by-step guidance — no confusion left behind.