Solving by substitution

The idea: use one equation to say what x is, then put that into the other equation. Now you have one equation with one unknown, which you already know how to solve.

Solve 7x – 15y = 2 and x + 2y = 3:

• Step 1: pick the easier equation and make one letter the subject: x + 2y = 3, so x = 3 – 2y.
• Step 2: put this into the other equation: 7(3 – 2y) – 15y = 2, so 21 – 29y = 2 and y = 19/29.
• Step 3: put y back into x = 3 – 2y: x = 3 – 38/29 = 49/29.
• Check in both equations.

Sometimes the letters disappear in Step 2. That tells you something:

Example: two rails are x + 2y – 4 = 0 and 2x + 4y – 12 = 0. From the first, x = 4 – 2y. Put it in: 2(4 – 2y) + 4y – 12 = 0 becomes –4 = 0. That is false, so the rails never cross.

Word problems: write it as it says. 'Is' becomes =, 'times' becomes ×, 'more' becomes +, 'less' becomes –. Let the letters stand for exactly what is asked.

For ages, make a small table: a column each for 'years ago', 'now' and 'years later'. Let the ages now be the letters. Seven years ago, Aftab was s – 7 and his daughter t – 7. In three years, they will be s + 3 and t + 3.