Teach Me Maths

Identities and Algebraic Arguments

Question 11 mark

Which of these is an identity?

Hint

An identity is true for every allowed value of its letter.

Worked solution

Which of these is an identity?

  1. 2x+5=11
  2. 3(x+2)=3x+6
  3. 7x-2
  4. x\geq4
  1. Expanding 3(x+2) gives 3x+6 for every x, so this equality is an identity.
  2. 2x+5=11 is true only when x=3; 7x-2 is an expression; x\geq4 is an inequality.

Question 21 mark

Which statement about 2x+3=7 is correct?

Hint

Solve the equation and distinguish one solution from an identity.

Worked solution

Which statement about 2x+3=7 is correct?

  1. It is true for every value of x.
  2. It is true only when x = 2.
  3. It is true only when x = 5.
  4. It is never true.
  1. 2x+3=7 gives 2x=4, so x=2.
  2. It is an equation with one solution, not an identity true for every x.

Question 31 mark

Find k so that 6x+2x=kx is an identity.

Hint

Collect the like terms on the left.

Worked solution

Find k so that 6x+2x=kx is an identity.

  1. 6x+2x=8x for every x. Therefore k=8.

Question 41 mark

Find c so that 4(x+3)=4x+c is an identity.

Hint

Multiply both terms in the bracket by 4.

Worked solution

Find c so that 4(x+3)=4x+c is an identity.

  1. 4(x+3)=4x+12. Matching the constant terms gives c=12.

Question 51 mark

Which reason correctly shows that 3(2x-5) and 6x-15 are equivalent?

Hint

A valid algebraic reason must work for every x.

Worked solution

Which reason correctly shows that 3(2x-5) and 6x-15 are equivalent?

  1. Multiply both 2x and −5 by 3.
  2. Multiply 2x by 3 and leave −5 unchanged.
  3. They have the same value when x = 0, so that proves it.
  4. Both expressions contain x.
  1. Distribute 3 to both terms: 3\times2x+3\times(-5)=6x-15.
  2. This equality holds for every x. Testing one value alone would not establish an identity.

Question 61 mark

n is an integer. Which expression is always even?

Hint

An even integer is an integer multiplied by 2.

Worked solution

n is an integer. Which expression is always even?

  1. 2n+1
  2. n+2
  3. 2n
  4. \frac n2
  1. 2n is twice an integer, so it is always even.
  2. 2n+1 is odd; n+2 can be odd; \frac n2 need not even be an integer.

Question 71 mark

n is an integer. Write an expression, in terms of n, for the next integer after n.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Think about how far apart two integers that follow each other are, such as 7 and 8.

Worked solution

n is an integer. Write an expression, in terms of n, for the next integer after n.

  1. Adding 1 gives the next integer: n+1.

Question 81 mark

A student says, “When you add two prime numbers, the answer is always an even number.” Which calculation shows that the student is wrong?

Hint

Both numbers in the calculation must be prime, but the answer must go against the claim.

Worked solution

A student says, “When you add two prime numbers, the answer is always an even number.” Which calculation shows that the student is wrong?

  1. 3+5=8
  2. 2+7=9
  3. 5+7=12
  4. 4+6=10
  1. 2 and 7 are both prime numbers, but 2+7=9 is odd. This one counterexample shows the claim is wrong.
  2. 3+5=8 and 5+7=12 agree with the claim, and 4 and 6 are not prime numbers.

Question 91 mark

Leo tests x^2=x with x=0 and gets a true statement. Why does this not prove an identity?

Hint

Think about what the word every requires.

Worked solution

Leo tests x^2=x with x=0 and gets a true statement. Why does this not prove an identity?

  1. An identity can never contain a square.
  2. One successful value does not show the equality works for every value.
  3. Zero cannot be substituted into algebra.
  4. All equations are identities.
  1. At x=0 the equality is true, but at x=2, x^2=4 and x=2.
  2. It fails at this value, so it is not an identity. One successful test was not a general proof.

Question 102 marks

a(2x+3)=10x+b is an identity. Find the constants.

(a)

Find a.

1 mark

(b)

Find b.

1 mark

Hint

Expand the left-hand side, then compare the x-terms and constant terms separately.

Worked solution

a(2x+3)=10x+b is an identity. Find the constants.

Part (a)

Find a.

  1. The x-term on the left is 2ax. Match coefficients: 2a=10, so a=5.

Part (b)

Find b.

  1. The constant term is 3a=3\times5=15, so b=15.

Question 111 mark

n is an integer. The next integer after n is n+1. Write a simplified expression for the sum of these two integers.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Add the two expressions and collect the n-terms.

Worked solution

n is an integer. The next integer after n is n+1. Write a simplified expression for the sum of these two integers.

  1. n+(n+1)=n+n+1=2n+1.
  2. This is one more than twice an integer, so the sum is odd.

Question 121 mark

Two odd integers are written as 2a+1 and 2b+1, where a and b are integers. Which is a valid general reason that their sum is even?

Hint

Look for an expression that covers all integers a and b.

Worked solution

Two odd integers are written as 2a+1 and 2b+1, where a and b are integers. Which is a valid general reason that their sum is even?

  1. The sum is 2(a+b+1), which is twice an integer.
  2. 3+5=8, so every possible sum is even.
  3. The sum is 2a+2b+1, which is even.
  4. Two odd numbers are always equal.
  1. (2a+1)+(2b+1)=2a+2b+2=2(a+b+1).
  2. a+b+1 is an integer, so the result is twice an integer and is even. A numerical example alone is not a general argument.

Question 133 marks

Every edge of the L-shape is horizontal or vertical. Its overall width is (3x+2) cm and its overall height is (2x+1) cm. The number x is a positive integer.

(a)

Write a simplified expression for its perimeter in centimetres.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Which statement is always true about the perimeter in centimetres?

1 mark

Hint

For this L-shape, combine the horizontal lengths and the vertical lengths.

Worked solution

Every edge of the L-shape is horizontal or vertical. Its overall width is (3x+2) cm and its overall height is (2x+1) cm. The number x is a positive integer.

Part (a)

Write a simplified expression for its perimeter in centimetres.

  1. The horizontal edges together total twice the overall width. The vertical edges together total twice the overall height.
  2. P=2(3x+2)+2(2x+1)=6x+4+4x+2=10x+6 cm.

Part (b)

Which statement is always true about the perimeter in centimetres?

  1. It is an odd integer.
  2. It is an even integer.
  3. It is a multiple of 10.
  4. It equals the area.
  1. 10x+6=2(5x+3). Since x is an integer, 5x+3 is an integer, so the perimeter is even.

Question 142 marks

A claim says, “For every positive integer n, the number 5n ends in the digit 5.”

(a)

Find the smallest positive integer n that shows the claim is wrong.

1 mark

(b)

Why is one example like this enough to show the claim is wrong?

1 mark

Hint

Start with the smallest allowed integers, then consider the word every.

Worked solution

A claim says, “For every positive integer n, the number 5n ends in the digit 5.”

Part (a)

Find the smallest positive integer n that shows the claim is wrong.

  1. For n=1, 5n=5, which does end in 5. For n=2, 5n=10, which does not.
  2. Therefore the smallest positive integer that shows the claim is wrong is n=2.

Part (b)

Why is one example like this enough to show the claim is wrong?

  1. It proves the claim works for all other integers.
  2. It shows the claim is not true for every positive integer.
  3. It shows no positive integer works.
  4. It shows 5 is not an integer.
  1. The claim says every positive integer works. One that does not work is enough to show that this universal claim is false.

Question 152 marks

n can be any integer. Find the smallest positive integer k for which 6n+k is always a multiple of 3.

Hint

The term involving n is already a multiple of 3. Also consider n = 0.

Worked solution

n can be any integer. Find the smallest positive integer k for which 6n+k is always a multiple of 3.

  1. 6n is always a multiple of 3. At n=0, the expression equals k, so k must also be a multiple of 3.
  2. The smallest positive multiple of 3 is 3. Then 6n+3=3(2n+1), which always works.

Question 16Challenge3 marks

2n, 2n+2, 2n+4 and 2n+6 are four even numbers, where n is an integer.

(a)

Their sum can be written as 4(\square). Write an expression for the box.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Which statement is true for every integer n?

1 mark

Hint

First add the four expressions, then examine the factor left after taking out 4.

Worked solution

2n, 2n+2, 2n+4 and 2n+6 are four even numbers, where n is an integer.

Part (a)

Their sum can be written as 4(\square). Write an expression for the box.

  1. The sum is 8n+12. Factor out 4: 8n+12=4(2n+3).
  2. The box is 2n+3.

Part (b)

Which statement is true for every integer n?

  1. The sum is divisible by 8.
  2. The sum is divisible by 4 but not by 8.
  3. The sum is odd.
  4. The sum is always positive.
  1. 2n+3 is odd, so 4(2n+3) is four times an odd integer.
  2. It is divisible by 4, but dividing it by 8 would give half an odd integer, not an integer.

Question 17Challenge3 marks

3(ax-2)+2(x+b)=17x+8 is an identity. Find the constants.

(a)

Find a.

2 marks

(b)

Find b.

1 mark

Hint

Expand both brackets, then match the coefficients and the constants.

Worked solution

3(ax-2)+2(x+b)=17x+8 is an identity. Find the constants.

Part (a)

Find a.

  1. The left side expands to 3ax-6+2x+2b. Its x-coefficient is 3a+2.
  2. 3a+2=17 gives 3a=15, so a=5.

Part (b)

Find b.

  1. Match constants: -6+2b=8. Thus 2b=14, so b=7.

Question 18Challenge4 marks

For any integer n, consider n(n+1)+(n+1)(n+2).

(a)

Expand and simplify this expression.

3 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Which is a valid reason that the expression is always even?

1 mark

Hint

Work with both products. A general argument must apply to every integer n.

Worked solution

For any integer n, consider n(n+1)+(n+1)(n+2).

Part (a)

Expand and simplify this expression.

  1. n(n+1)=n^2+n.
  2. (n+1)(n+2)=n^2+3n+2.
  3. Adding gives 2n^2+4n+2.

Part (b)

Which is a valid reason that the expression is always even?

  1. It is 2(n^2+2n+1), and the bracket is an integer.
  2. It is a sum of two numbers, so it must be even.
  3. It contains n^2, so it must be even.
  4. It equals 2 when n=0, which proves every case.
  1. 2n^2+4n+2=2(n^2+2n+1).
  2. The bracket is an integer for every integer n, so the expression is always twice an integer.

Question 19Challenge3 marks

A rectangular frame has outer width (4n+2) cm and outer height (2n+1) cm, where n is a positive integer. The border is 1 cm wide all the way round.

(a)

Find a simplified expression for the inner perimeter in centimetres.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Which statement is always true about this inner perimeter in centimetres?

1 mark

Hint

The inner dimensions each lose a border at both ends. Then use the algebraic form of the perimeter.

Worked solution

A rectangular frame has outer width (4n+2) cm and outer height (2n+1) cm, where n is a positive integer. The border is 1 cm wide all the way round.

Part (a)

Find a simplified expression for the inner perimeter in centimetres.

  1. Subtract two border widths from each outer dimension: inner width 4n+2-2=4n cm; inner height 2n+1-2=2n-1 cm.
  2. The inner perimeter is 2(4n)+2(2n-1)=12n-2 cm.

Part (b)

Which statement is always true about this inner perimeter in centimetres?

  1. It is divisible by 4.
  2. It is odd.
  3. It is even but not divisible by 4.
  4. It is always 12.
  1. 12n-2=2(6n-1), and 6n-1 is odd.
  2. The perimeter is twice an odd integer, so it is even but not divisible by 4.

Question 20Challenge4 marks

Tom claims that (n+3)^2 and n^2+9 are identical expressions.

(a)

Expand and simplify the difference (n+3)^2-(n^2+9).

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Find the smallest positive integer n for which this difference is greater than 30.

2 marks

Hint

Expand the square as a product of two identical brackets before subtracting.

Worked solution

Tom claims that (n+3)^2 and n^2+9 are identical expressions.

Part (a)

Expand and simplify the difference (n+3)^2-(n^2+9).

  1. (n+3)^2=n^2+6n+9.
  2. Subtract n^2+9: the difference is n^2+6n+9-n^2-9=6n. It is not always zero, so the claim is false.

Part (b)

Find the smallest positive integer n for which this difference is greater than 30.

  1. The difference is 6n, so 6n>30. This gives n>5.
  2. The smallest positive integer allowed is 6. The difference is then 36.

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