Simultaneous Equations Solver

Simultaneous Equations Guide

What's the key thing to understand about 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 ✓.

What do I need to know about 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 ✓.

What do I need to know about 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.

What's the key thing to understand about 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.