Algebra · Simultaneous equations
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Substitution and real-world simultaneous equations
Why substitution collapses two equations into one, the bracket that guards every substitution, and how to translate two-fact word problems (prices, sums, perimeters, intersections) into a solvable pair.
Algebra · Simultaneous equations
Substitution and real-world simultaneous equations
Why substitution collapses two equations into one, the bracket that guards every substitution, and how to translate two-fact word problems (prices, sums, perimeters, intersections) into a solvable pair.
Why it works
Substitution trades one unknown for the other. If one equation already isolates a variable — — then wherever the other equation says , you may write instead:Two equations became one equation in one unknown — the familiar kind. Substitution shines exactly when a formula-form equation () is already on the table; elimination shines when both equations are in shape.
The bracket is non-negotiable. Substituting into gives — the 3 multiplies the WHOLE of . Dropping the bracket () silently loses the and wrecks everything after. And always substitute into the OTHER equation — feeding back into itself just proves .
Word problems: two facts, two equations. Let the letters be NUMBERS (costs, ages, lengths — with units), then translate each sentence:
- "3 adult and 2 child tickets cost £24.30" →
- "2 adult and 3 child tickets cost £21.20" →
An intersection of two lines IS a simultaneous solution. The lines and cross where both equations hold at once: , so , — the point . This is the picture behind every pair of simultaneous equations: two lines, one crossing point.
Sense-check the context. Prices come out in sensible money, lengths positive, ages human. A camel-sized ticket price means a slip upstream — recheck the subtraction, not the world.