Brahmagupta’s formula
Ax + B = Cx + D has the solution x = (D − B) ÷ (A − C).
Four numbers, one answer
Look at 5x + 4 = 3x + 8. The numbers are 5, 4, 3 and 8. Brahmagupta wondered: can the four numbers alone, put through some subtractions and a division, hand us x directly?
What do you think x is in 5x + 4 = 3x + 8?
What this lesson covers
The idea
An equation of the form Ax + B = Cx + D, where A, B, C and D are numbers, has the solution x = (D – B) ÷ (A – C).
Four numbers, one answer
Look at 5x + 4 = 3x + 8. The numbers are 5, 4, 3 and 8. Brahmagupta wondered: can the four numbers alone, put through some subtractions and a division, hand us *x* directly?
What do you think *x* is in 5x + 4 = 3x + 8?
- x = 2
- x = 4
- x = 12
Build the formula, then test it
Call the numbers A, B, C, D: Ax + B = Cx + D. Fill the four slots to make a formula for *x*, then test it on the equations. It has to pass every test.
Why the formula works
Do the same to both sides. Take away B and take away Cx. What is left is (A − C)x = D − B. Then divide both sides by A − C.
An equation like the book’s 650m + 4000 = 500m + 5050 becomes just a calculation: m = (5050 − 4000) ÷ (650 − 500) = 7. Brahmagupta’s formula is a recipe that anyone can follow.
An equation of the form Ax + B = Cx + D has the solution x = (D − B) ÷ (A − C).
- Step | In letters | Example
- start | Ax + B = Cx + D | 5x + 4 = 3x + 8
- take B away | Ax = Cx + D − B | 5x = 3x + 4
- take Cx away | (A − C)x = D − B | 2x = 4
- divide by A − C | x = (D − B) ÷ (A − C) | x = 4 ÷ 2 = 2
Notes
Brahmagupta’s formula: an equation Ax + B = Cx + D has the solution x = (D − B) ÷ (A − C). For 5x + 4 = 3x + 8: x = (8 − 4) ÷ (5 − 3) = 2.
Check yourself
Use the formula to solve 7x + 1 = 2x + 16.
Answer: 3
x = (16 − 1) ÷ (7 − 2) = 15 ÷ 5 = 3. Check: 7 × 3 + 1 = 22 and 2 × 3 + 16 = 22.
Use the formula to solve 4x + 8 = 2x + 2.
Answer: -3
x = (2 − 8) ÷ (4 − 2) = −6 ÷ 2 = −3. Check: 4 × (−3) + 8 = −4 and 2 × (−3) + 2 = −4.
A student writes (B − D) ÷ (A − C) for 5x + 4 = 3x + 8. She gets x = −2. What went wrong?
What happens to the formula for 5x + 3 = 5x + 8?
- Nothing: x = −2 is correct.. Put it in: 5 × (−2) + 4 = −6, but 3 × (−2) + 8 = 2. They are not equal.
- The top should be D − B. With x = −2 the left side is −6 and the right side is 2, so they are not equal. — correct. Yes! (B − D) has the wrong sign. (D − B) ÷ (A − C) = 4 ÷ 2 = 2.
- A − C should have been A + C.. The bottom A − C = 2 is right. It is the top that has the wrong sign.
- x = (8 − 3) ÷ 5 = 1, so x = 1.. Put x = 1 in: 5 + 3 = 8, but 5 + 8 = 13. Not equal. The bottom of the formula is A − C, not A.
- x = 0, because both sides start with 5x.. Put x = 0 in: 3 on the left and 8 on the right. Not equal.
- A − C = 5 − 5 = 0, so we cannot divide. No value of x works, since 5x + 3 is always 5 less than 5x + 8. — correct. Yes! The formula needs A and C to be different.