Simultaneous equations

Last time

Before we start: last time, in Quadratic equations, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Two clues, two unknowns

Two adults and three children pay thirty-one dollars. One adult and two children pay eighteen. What does each ticket cost? One clue is not enough, but two clues together are. Today you solve pairs of equations by elimination and by substitution, turn word problems into equations, and meet a straight line cutting a curve.

What simultaneous means

Simultaneous equations are two equations that are true at the same time, for the same values of the unknowns. On a graph, each equation is a straight line, and the solution is the point where the lines cross. There are two main methods. Elimination removes one unknown by adding or subtracting. Substitution replaces one unknown.

Elimination: matching coefficients

Take two x plus three y equals twelve, and four x minus three y equals six. The y terms are three y and minus three y: they will cancel if you add. Add the equations. Two x plus four x is six x. The y terms cancel. Twelve plus six is eighteen. So x is three. Substitute into the first equation: six plus three y equals twelve. Three y is six, so y is two. Check in the second: twelve minus six is six.

Same signs: subtract

If the matching terms have the same sign, subtract the equations. If they have opposite signs, add. A memory trick: same signs subtract, SSS. The terms to remove are the ones with the same size and the same sign. Always write the new equation before solving, and substitute the answer back to find the second unknown.