Simultaneous Equations Solver

Simultaneous Equations Guide

The Elimination Method

Multiply one or both equations to make the coefficient of one variable equal. Then add or subtract the equations to eliminate that variable. Example: 2x + 3y = 12 and 4x + y = 10. Multiply equation 2 by 3: 12x + 3y = 30. Subtract equation 1: 10x = 18, so x = 1.8. Substitute back: 2(1.8) + 3y = 12 → 3y = 8.4 → y = 2.8. Verify in equation 2: 4(1.8) + 2.8 = 7.2 + 2.8 = 10 ✓.

The Substitution Method

Rearrange one equation to express one variable in terms of the other, then substitute into the second equation. Best used when one equation has a coefficient of 1 or -1 (easy to rearrange). Example: y = 3 - x and 2x + 5y = 9. Substitute: 2x + 5(3-x) = 9 → 2x + 15 - 5x = 9 → -3x = -6 → x = 2. Then y = 3 - 2 = 1. Verify in equation 2: 2(2) + 5(1) = 4 + 5 = 9 ✓.

Graphical Interpretation

Two linear equations each represent a straight line on a graph. The solution (x, y) is the point where the lines intersect. If the lines are parallel (same gradient, different y-intercept), there is no solution. If the lines are identical (same gradient, same y-intercept), there are infinite solutions. For exam questions, always check for 'no solution' (inconsistent) or 'infinite solutions' (dependent) cases before calculating.

Real-World Applications

Simultaneous equations model any situation with two conditions constraining two unknowns. Examples: finding break-even point (total cost = total revenue); mixture problems (how much of each ingredient to achieve a target concentration); distance problems (two objects moving at different speeds); finance (splitting costs between two accounts with different interest rates). Setting up the equations correctly from a word problem is often harder than solving them.