Rearranging a formula.

Step 1 of 4

A formula as a line

A plumber charges a call-out fee of $3$⁠ shekels plus $2$⁠ shekels for every hour of work (small on purpose, so they fit on the board). The cost $y$⁠ of $x$⁠ hours is

$y = 2x + 3$⁠

The formula quickly answers “how much will it cost?”: substitute $x$⁠ and work it out. The opposite question, “how many hours can a budget of $11$⁠ buy?”, is an equation: $2x + 3 = 11$⁠. On the board the budget is the pink line, and the answer is where it meets the cost line.

Step 2 of 4

Isolate x

Instead of solving a new equation for every budget, you can solve once, with the letter $y$⁠ in place of a number. The steps are exactly the same: subtract $3$⁠ from both sides and divide both sides by $2$⁠. The result is a new formula that takes a budget and gives back hours.

Change Budget and watch the meeting point slide along the line.

Step 3 of 4

Other formulas

The same method works in any formula, even one with several letters. The perimeter of a rectangle with sides $a$⁠ and $b$⁠ is

$P = 2a + 2b$⁠

To isolate $b$⁠, treat all the other letters as if they were numbers: whatever is added to $2b$⁠ moves to the other side, and whatever multiplies $b$⁠ is divided out.

Step 4 of 4

From hours to budget

Now the other way round: you know how many hours you need, and you look for the budget. Price per hour and Call-out fee are locked, which leaves only Budget.