Linear equations and input-output

When we set a linear polynomial equal to a number, we get a linear equation.

Example 6. Two numbers add to 64, and one is 10 more than the other. Call the smaller x, so the larger is x + 10. Then x + (x + 10) = 64, so 2x + 10 = 64, 2x = 54 and x = 27. The numbers are 27 and 37.

A polynomial is a machine: put a number in, get a number out. For 2x + 3:

• input 4 → 2 × 4 + 3 = 11;
• input –6 → 2 × (–6) + 3 = –9.

We say 2x + 3 is a function of x.

The wire-rectangle area 10x – x² is a function of x too: input 6 → 60 – 36 = 24 cm². 2x + 3 is a linear function; 10x – x² is a quadratic function.

For any word problem:

• name the unknown;
• write the expression;
• set it equal to what you are told;
• solve;
• check the answer in the story. (If you get stuck, you can also guess, test and fix: try a number, see if it fits, adjust.)