Line From Start and Rate: y = mx + b, Two Points and Break-Even

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Opens on Chapter 13’s example: an electrician’s $45 call-out plus $60 an hour, against another at $75 an hour and no call-out fee.

Line 1: y = mx + b

Rate m

Start b

Line 2 (optional)

Rate m

Start b

Work out y at x = one or more

Or: the line through two points

x₁

y₁

x₂

y₂

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Enter your numbers, then press Work it out.

How it works

In y = mx + b, b is where the line starts (the value at x = 0) and m is how much y goes up for each 1 in x. From two points, m = (y₂ − y₁) ÷ (x₂ − x₁) and then b = y₁ − m × x₁. A line reaches zero at x = −b ÷ m. Two lines cross where mx + b is the same for both: m₁x + b₁ = m₂x + b₂ gives x = (b₂ − b₁) ÷ (m₁ − m₂); if the rates match, the lines are parallel and never meet.

From the book, Chapter 13. The first electrician charges $165 for 2 hours and $345 for 5; from those two points m = 60 and b = 45. Against $75 an hour the two cost the same at 3 hours ($225). Sanjay’s price and cost meet at 50 sheets ($32), and his profit at 200 sheets is $45. The candle falls −0.75 in an hour and burns out at 16 hours. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.