7.3 Introduction to Algebra

If $a = 6$, find a value of $b$ so that $3a + b = 21$. Justify your answer.

Which of these is the correct algebraic convention: $n \times n$ or $n^2$? Explain why the other form is wrong.

Write $$11x$ in two different ways as a sum of like terms. Show each is equivalent to the original.

Make up an equation whose solution is $x = 7$. Solve it to check.

A number of chairs $n$ is arranged in 5 equal rows. Write an equation for this and solve it to find $n$.

Expand $5(x + 2)$ and show that your answer gives the same value as the original expression when $x = 3$.

Solve $3x + 4 = 19$ and check your answer by substituting back.

Write two different expressions that both describe "a number $n$ multiplied by 6, then subtract 5". Explain why they are the same.

Decide: are $8y$ and $3y$ like terms? Simplify $8y - 3y$ and justify your step.

In the expression $4x + 9$, identify the coefficient of $x$ and the constant. Explain what each part tells you about the expression.

Little puzzle: what number $a$ makes $a \times 7 = 0$? How did you know?

Find the value of $2a + 3b$ when $a = 5$ and $b = 2$.

Simplify: $7x + 4y + 3x - 2y$.

Expand and simplify: $3(x + 5) + 7$.

Solve: $5x - 2 = 23$.

Tickets to a show cost $\pounds {p}$ each. 7 friends buy tickets. Write an expression for the total cost, then find the cost when $p = 6$.

Find the value of $2x^2$ when $x = 5$.

Simplify: $8a + 7b + 1a - 4b$.

A rectangle has length $(x + 5)$ cm and width $6$ cm. Write a simplified expression for its perimeter.

Expand the bracket: $3(4x - 7)$.

Solve: $\dfrac{x}{4} + 3 = 8$.

Another little puzzle: what number $a$ makes $5(2 - a) = 0$? Try to spot the answer without doing lots of working.

Simplify: $6x + 4x - 3x$.

Find the value of $3a^2 + 2a$ when $a = -3$.

Expand and simplify: $4(x + 3) + 3(x + 2)$.

Solve: $5x + 9 = -1$.

A bag of apples costs ${p}p$ pence and a bag of oranges costs ${q}p$ pence. Write and simplify an expression for the total cost of 5 bags of apples and 2 bags of oranges.

Work out $4a - 3b$ when $a = 2$ and $b = -5$.

Solve: $6x + 2 = 2x + 14$.

A rectangle has length $(2x + 2)$ cm and width $(x + 4)$ cm. Its perimeter is 36 cm. Find $x$.

Expand and simplify: $4(3x - 2) - 6x$.

If $n = 2^2 \times 3$, find the value of $3n - 4$.

I think of a number. I double it, then subtract 5. The result is -9. Find the number.

Three consecutive integers have a sum of 63. Find the three integers and write an equation to show your method.

Solve: $\dfrac{2x + 4}{3} = 6$.

The expression $5n - 3$ is evaluated for two consecutive negative integers. Show that the difference between the two results is always 5, whatever negative integers you choose.

A mobile phone plan charges a fixed monthly fee of $\pounds 15$ plus $\pounds 4$ per GB of data used. Write a formula for the monthly cost $C$ in terms of $g$ (GB used). Use it to find how many GB were used if the bill was $\pounds 51$.

Alice thinks of a prime number $p$. She computes $p^2 - p$. Show algebraically that the result is always even. Verify with $p = 7$.

Two numbers, $x$ and $y$, satisfy: $x + y = 20$ and $x - y = 6$. Find $x$ and $y$.

The perimeter of a triangle is $46$ cm. One side is $(3x + 2)$ cm, a second side is $(2x - 3)$ cm, and the third side is $7$ cm. Find $x$ and the length of each side.

Explain why $6(x + 4)$ and $6x + 24$ are equal. What must 24 equal? Now expand $3(x - 5)$ and simplify $3(x - 5) + 7x$.

A number machine applies the rule "$\times 4$, then $- 10$". An input gives an output of $6$. Find the input. Then find the input that gives an output equal to the original input.