Solving ax + b = c: The Pattern Behind Countless Word Problems
An enormous number of practical problems reduce to one shape: something per unit, plus a fixed amount, equals a total. That is ax plus b equals c, and the Linear Equation Solver solves it for x.
Two steps, always the same order
Subtract b from both sides, then divide both sides by a. So x equals (c minus b) divided by a.
For 5x plus 12 equals 47: subtract 12 to get 5x equals 35, divide by 5 to get x equals 7.
The order matters, and it is the reverse of the order the equation builds up in. The equation multiplies by a and then adds b, so solving undoes the addition first and the multiplication second - last operation first, like taking off shoes before socks.
The hard part is writing it down
Almost nobody struggles with the algebra. The difficulty is turning words into the equation, and the trick is to identify the three parts by what they do.
a is the rate - the cost per item, the speed, the amount per hour, the charge per unit. b is the fixed part - the call-out fee, the deposit, the base charge, the starting value. c is the total you know.
"A plumber charges 60 to come out plus 45 an hour, and the bill was 240. How long were they there?" Rate is 45, fixed part is 60, total is 240. So 45x plus 60 equals 240, and x is 4.
"A phone plan costs 15 a month plus 0.10 a minute, and the bill was 27." Rate 0.10, fixed 15, total 27, so x is 120 minutes.
Taxi fares, utility bills, break-even volumes, mileage, delivery charges, savings targets - all the same equation with different words. The Break-Even Point Calculator is this equation applied to a business.
Check by substituting back
Put your answer into the original equation and see whether it balances. 45 times 4 plus 60 is 240. It does.
This takes five seconds and catches nearly every arithmetic slip, which makes it the most worthwhile habit in solving anything.
The cases worth watching
A negative a means dividing by a negative, which flips the sign of the answer - and if it is an inequality rather than an equation, dividing by a negative also reverses the inequality sign.
a equals zero means there is no x in the equation at all. Either it is always true or never true, and neither has a solution for x.
An answer that makes no sense in context - a negative number of people, 2.5 hours when the plumber charges whole hours. The algebra is right and the model needs rounding or rethinking, which is a judgement rather than a calculation.
Common questions
What if x appears on both sides?
Collect the x terms on one side first by subtracting, then it is the same two steps.
What if there are brackets?
Expand them first, then collect like terms.
What if the equation has a squared term?
Then it is quadratic, and the Quadratic Equation Solver is the right tool.
Got an equation to solve?