इन सवालों के जवाब देकर एनर्जी कमाएं, फिर मछली पकड़ें और घूमें। कोई अकाउंट नहीं चाहिए।
शिक्षकों के लिए: Rearranging formulae and algebraic proof (Mathematics, Algebra) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके शिक्षार्थी लाइव गेम की तरह खेलें।
A formula shows a relationship between variables, e.g. v = u + at (v = final velocity, u = initial velocity, a = acceleration, t = time).
The subject of a formula is the variable on its own on one side, e.g. in A = \frac{1}{2} b h, A is the subject.
Rearranging (or changing the subject) means rewriting the formula so a different variable becomes the subject.
An equation states equality; an identity is an equation true for every permitted value. 3x + 2 = 11 has the particular solution x = 3.
In an identity, use the symbol \equiv to show it is always true.
Algebraic proof uses algebra to show a statement is always true, not just for a few examples.
Rearranging Formulae: The Golden Rule
Whatever you do to one side of the formula, do exactly the same to the other side.
Use inverse operations, taking care with restrictions: division needs a nonzero divisor, and reversing a square generally introduces positive and negative roots.
Work in the reverse order of operations (like undoing a function machine).
Example: make r the subject of C = 2\π r: divide both sides by 2\π to get r = \frac{C}{2\π}.
Example: make h the subject of A = \frac{1}{2} b h: multiply both sides by 2, then divide by b: h = \frac{2A}{b}.
Always state the new subject clearly, e.g. t = \frac{v - u}{a} from v = u + at.
Dealing with Roots and Powers
If x² = A for A ≥ 0, then x = ±√A. A length, radius or nonnegative speed uses the nonnegative branch.
Example: A = \π r2 → r = \√{\frac{A}{\π}} (take the positive root unless context says otherwise).
To remove a cube, take the cube root: V = x3 → x = \√[3]{V}.
If the subject is inside a fraction, first multiply to clear the denominator.
Example: P = \frac{F}{A} → F = PA and A = \frac{F}{P}.
Be careful: \√{a + b} is not the same as \√{a} + \√{b}.
Subject Appears More Than Once (Higher Tier)
If the new subject appears in more than one term, collect those terms on one side of the formula.
Then factorise to take the subject out as a common factor.
Example: make x the subject of ax + b = cx + d: bring cx to the left: ax - cx = d - b.
Factorise: x(a - c) = d - b, then divide: x = \frac{d - b}{a - c}.
Example: make t the subject of s = ut + \frac{1}{2} a t2: this is quadratic in t, so use factorising or the quadratic formula (Higher).
Always check that you have not divided by zero (e.g. a - c \neq 0).
Equations vs Identities
An equation may hold for particular values, no values, or every permitted value. An identity is the case that holds for every permitted value.
An identity is true for every value of the variable (e.g. 3(x + 2) \equiv 3x + 6).
The symbol ≡ emphasises an identity. An equals sign can also appear in a true identity.
To prove an identity, expand and simplify one side until it matches the other side.
Example: show (x + 3)2 - 9 \equiv x(x + 6): expand left to x2 + 6x + 9 - 9 = x2 + 6x = x(x + 6).
You cannot prove an identity by substituting numbers; you must use algebra.
Algebraic Proof: Representing Numbers
Use 2n to represent an even number (n is an integer).
Use 2n + 1 (or 2n - 1) for an odd number.
Consecutive integers can be written as n, n+1, n+2 (or n, n+1, n+2, \ldots).
Consecutive even numbers: 2n, 2n+2, 2n+4.
Consecutive odd numbers: 2n+1, 2n+3, 2n+5.
When proving statements about odd/even numbers, always state that n is an integer.
Constructing Algebraic Proofs
Start with a general form (e.g. 2n+1 for an odd number) and manipulate algebraically.
To prove a result is a multiple of 3, show it can be written as 3 \times (some expression).
To prove a result is even, show it can be written as 2 \times (some integer expression).
Example: prove the sum of three consecutive integers is a multiple of 3: n + (n+1) + (n+2) = 3n + 3 = 3(n+1).
Example: prove the product of two consecutive even numbers is divisible by 8: let the numbers be 2n and 2n+2; product = 4n(n+1), and since n(n+1) is even, product is divisible by 8.
Always conclude with a clear statement, e.g. 'Therefore the sum is always a multiple of 3.'
Common Mistakes to Avoid
Do not 'move' terms without applying the inverse operation to the whole side.
When taking a root, take the root of the entire expression, not individual terms.
If the subject appears twice, do not stop after collecting terms; factorise to isolate it.
Do not use a few numerical examples as proof; algebra must be general.
Do not use the same letter for two different quantities in a proof.
Always check your final formula by substituting simple numbers.
Typical Exam Question Types
Make a stated variable the subject of a formula (e.g. make x the subject of y = \frac{ax + b}{c}).
Rearrange a scientific formula and then substitute values (e.g. F = ma to find a).
Prove that the sum of three consecutive integers is always a multiple of 3.
Show that a given expression is always even or always positive (e.g. (n+1)2 - n2 is always odd).
Prove an algebraic identity by expanding and simplifying.
For Higher tier, handle formulae where the subject appears more than once.
स्लाइड्स
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फ्री प्रीव्यू — 61 में से 8 सवाल। सभी देखने के लिए साइन अप करें।
1.Every algebraic equation is an identity.
Easy
True or false?
2.Rearrange the formula v = u + at to make t the subject. Assume a ≠ 0.
Easy
At = (v - u) / a
Bt = (v + u) / a
Ct = v - u / a
Dt = (u - v) / a
3.Rearrange the formula A = 1/2 × b × h to make h the subject. Assume b ≠ 0.
Medium
Ah = 2A / b
Bh = A / (2b)
Ch = 2A × b
Dh = (A - b) / 2
4.Rearrange the formula P = F / A to make A the subject. Assume P ≠ 0.
Medium
AA = F / P
BA = P / F
CA = F × P
DA = F - P
5.Rearrange the formula y = (3x - 1) / 2 to make x the subject.
Medium
Ax = (2y + 1) / 3
Bx = (2y - 1) / 3
Cx = (2y + 1) × 3
Dx = (2y - 1) × 3
6.Make t the subject of the formula s = ut + 1/2 a t2. Assume a ≠ 0 and real roots exist.
Medium
At = (-u ± √(u2 + 2as)) / a
Bt = (-u ± √(u2 - 2as)) / a
Ct = (u ± √(u2 + 2as)) / a
Dt = (-u ± √(u2 + 2as)) / (2a)
7.Given the formula a = (b + c) / (b - c), make b the subject. Assume a ≠ 1 and b ≠ c.
Medium
Ab = c(a + 1) / (a - 1)
Bb = c(a - 1) / (a + 1)
Cb = c(1 + a) / (1 - a)
Db = c(a + 1) / (1 - a)
8.Which of the following are algebraic identities? (select all that apply)
Medium
Ax2 - y2 = (x + y)(x - y)
Bx2 + 2xy + y2 = (x + y)2
Cx2 + y2 = (x + y)2
D2(x + 3) = 2x + 6
और सवाल
9.The expression 2n + 1 always represents an odd number for integer n.
Easy
True or false?
10.Which of the following is the correct first step to prove that (n + 1)2 - n2 = 2n + 1?
Medium
AExpand (n + 1)2 to n2 + 2n + 1
BSubstitute n = 1
CSubtract n2 from both sides
DFactorise n2
11.To prove an identity, it is sufficient to check that it works for a few numerical values.
Easy
True or false?
12.Which of the following is a correct algebraic proof that the product of two consecutive odd numbers is odd?
Medium
ALet the numbers be 2n+1 and 2n+3. Product = (2n+1)(2n+3) = 4n2 + 8n + 3 = 2(2n2 + 4n + 1) + 1, which is odd.
BLet the numbers be 3 and 5. Product = 15, which is odd.
CLet the numbers be n and n+2. Product = n2 + 2n = n(n+2), which may be even or odd.
DLet the numbers be 2n+1 and 2n+1. Product = (2n+1)2 = 4n2 + 4n + 1, which is odd.
13.Rearrange the formula 1/u + 1/v = 1/f to make u the subject. Assume u, v and f are nonzero and v ≠ f.
Medium
Au = fv / (v - f)
Bu = fv / (v + f)
Cu = (v - f) / fv
Du = (v + f) / fv
14.Match each algebraic expression to its equivalent identity.
Medium
(x + y)2
x2 - y2
(x - y)2
x2 + 2xy + y2
(x + y)(x - y)
x2 - 2xy + y2
15.Put the following steps in order to rearrange the formula y = (3x + 5) / 2 to make x the subject.
Medium
Multiply both sides by 2: 2y = 3x + 5
Subtract 5 from both sides: 2y - 5 = 3x
Divide both sides by 3: x = (2y - 5) / 3
16.The area of a trapezium is A = ½(a + b)h. For a + b ≠ 0, rearrange the formula to make h the subject.
Medium
Ah = 2A / (a + b)
Bh = A / (2(a + b))
Ch = 2A × (a + b)
Dh = (a + b) / (2A)
17.Match each formula to its correct rearrangement to make x the subject.
Medium
y = 3x + 2
y = 5 - 2x
y = x/4 + 1
x = (y - 2)/3
x = (5 - y)/2
x = 4(y - 1)
18.Put the following steps in order to rearrange the formula v2 = u2 + 2as to make s the subject.
Medium
Subtract u2 from both sides: v2 - u2 = 2as
Divide both sides by 2a: s = (v2 - u2)/(2a)
Write the formula: v2 = u2 + 2as
19.Which of the following are correct rearrangements to make x the subject? (Select all that apply)
Medium
AFrom y = 2x + 3, x = (y - 3)/2
BFrom y = 2x + 3, x = y/2 - 3
CFrom y = (x + 1)/5, x = 5y - 1
DFrom y = (x + 1)/5, x = 5(y - 1)
EFrom y = 4 - 3x, x = (4 - y)/3
20.The statement (x + 1)2 = x2 + 2x + 1 is an identity.
Easy
True or false?
21.The identity (x + a)² ≡ x² + 6x + b holds for every real x. Find a and b.
Hard
Aa = 3, b = 9
Ba = 3, b = 6
Ca = 6, b = 36
Da = -3, b = 9
22.Which of the following are correct algebraic proofs that the sum of any two odd numbers is even? (Select all that apply)
Hard
ALet the numbers be 2n+1 and 2m+1. Their sum is 2n+2m+2 = 2(n+m+1), which is even.
BLet the numbers be 2n+1 and 2n+1. Their sum is 4n+2 = 2(2n+1), which is even.
CLet the numbers be 2n+1 and 2m+1. Their sum is 2(n+m+1), which is even.
DCheck with 3+5=8, which is even, so it must always be even.
23.Which of the following is the correct rearrangement of the formula y = (ax + b)/(cx + d) to make x the subject? Assume cx + d ≠ 0 and cy ≠ a.
Hard
Ax = (b - dy)/(cy + a)
Bx = (dy - b)/(a - cy)
Cx = (b + dy)/(cy - a)
Dx = (dy - b)/(cy - a)
24.Rearrange the formula C = 2πr to make r the subject. Which of the following is correct?
Medium
Ar = C - 2π
Br = C / (2π)
Cr = 2π / C
Dr = 2πC
25.The statement 3(x + 2) = 3x + 6 is an identity.
Easy
True or false?
26.Put the following steps in order to rearrange the formula y = (2x + 1)/3 to make x the subject.
Medium
Multiply both sides by 3: 3y = 2x + 1
Subtract 1 from both sides: 3y - 1 = 2x
Divide both sides by 2: x = (3y - 1)/2
27.Rearrange the formula y = (ax + b)/(cx + d) to make x the subject. Which of the following is correct? Assume cx + d ≠ 0 and cy ≠ a.
Medium
Ax = (dy - b)/(a - cy)
Bx = (dy + b)/(a - cy)
Cx = (dy - b)/(a + cy)
Dx = (dy + b)/(a + cy)
28.The formula for the volume V of a cylinder is V = πr2h. Rearrange to make h the subject, then find h when V = 100π cm3 and r = 5 cm.
Medium
Ah = 4 cm
Bh = 20 cm
Ch = 2 cm
Dh = 100 cm
29.The formula for the period T of a pendulum is T = 2π√(L/g), where L is length and g is gravitational acceleration. Which of the following are correct rearrangements to make L the subject? (Select all that apply)
Hard
AL = g(T/(2π))2
BL = gT2/(4π2)
CL = (gT2)/(4π2)
DL = gT/(2π)
30.The statement (n+2)2 - n2 = 4n + 4 is an identity.
Medium
True or false?
31.Put the following steps in order to rearrange the formula y = (3x + 5)/2 to make x the subject.
Medium
Multiply both sides by 2: 2y = 3x + 5
Subtract 5 from both sides: 2y - 5 = 3x
Divide both sides by 3: x = (2y - 5)/3
32.Kinetic energy E = ½mv², where m > 0 and v is nonnegative speed. Make v the subject.
Hard
Av = √(2E/m)
Bv = 2E/m
Cv = √(E/(2m))
Dv = (2E/m)2
33.The expression 2n + 2 always represents an even number for integer n.
Easy
True or false?
34.An identity holds for every permitted value; an equation that is not an identity can hold only for certain values.
Easy
True or false?
35.The expression 2n+1 always represents an odd number for integer n.
Easy
True or false?
36.Put the following steps in order to rearrange the formula y = (5x - 2)/3 to make x the subject.
Medium
Multiply both sides by 3
Add 2 to both sides
Divide both sides by 5
37.Which of the following are correct algebraic proofs that the sum of three consecutive integers is a multiple of 3? (Select all that apply)
Medium
ALet the integers be n, n + 1, n + 2. Their sum is 3n + 3 = 3(n + 1), which is a multiple of 3.
BLet the integers be n - 1, n, n + 1. Their sum is 3n, which is a multiple of 3.
