7.3 Introduction to Algebra

If $a = 4$, find a value of $b$ so that $3a + b = 15$. Justify your answer.

Which of these is the correct algebraic convention: $n3$ or $3n$? Explain why the other form is wrong.

Write $$8x$ in two different ways as a sum of like terms. Show each is equivalent to the original.

Make up an equation whose solution is $x = 8$. Solve it to check.

A number of chairs $n$ is arranged in 4 equal rows. Write an equation for this and solve it to find $n$.

Expand $3(x + 4)$ and show that your answer gives the same value as the original expression when $x = 2$.

Solve $2x + 5 = 17$ and check your answer by substituting back.

Write two different expressions that both describe "a number $n$ multiplied by 4, then subtract 3". Explain why they are the same.

Decide: are $9y$ and $4y$ like terms? Simplify $9y - 4y$ and justify your step.

In the expression $7x + 3$, identify the coefficient of $x$ and the constant. Explain what each part tells you about the expression.

Little puzzle: what number $a$ makes $a \times 5 = 0$? How did you know?

Find the value of $2a + 3b$ when $a = 3$ and $b = 4$.

Simplify: $5x + 3y + 2x - 1y$.

Expand and simplify: $4(x + 3) + 5$.

Solve: $4x - 3 = 21$.

Tickets to a show cost $\pounds {p}$ each. 5 friends buy tickets. Write an expression for the total cost, then find the cost when $p = 8$.

Find the value of $3x^2$ when $x = 4$.

Simplify: $6a + 5b + 2a - 3b$.

A rectangle has length $(x + 3)$ cm and width $4$ cm. Write a simplified expression for its perimeter.

Expand the bracket: $4(2x - 5)$.

Solve: $\dfrac{x}{3} + 4 = 9$.

Another little puzzle: what number $a$ makes $3(4 - a) = 0$? Try to spot the answer without doing lots of working.

Simplify: $3x + 5x - 2x$.

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

Expand and simplify: $3(x + 2) + 2(x + 5)$.

Solve: $5x + 12 = 2$.

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 3 bags of apples and 4 bags of oranges.

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

Solve: $5x + 3 = 2x + 15$.

A rectangle has length $(2x + 3)$ cm and width $(x + 5)$ cm. Its perimeter is 40 cm. Find $x$.

Expand and simplify: $3(2x - 4) - 5x$.

If $n = 2^2 \times 5$, find the value of $3n - 7$.

I think of a number. I double it, then subtract 7. The result is -3. Find the number.

Three consecutive integers have a sum of 48. Find the three integers and write an equation to show your method.

Solve: $\dfrac{3x + 6}{2} = 9$.

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 $\pounds 12$ plus $\pounds 3$ 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 $\pounds 33$.

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: $x + y = 14$ and $x - y = 4$. Find $x$ and $y$.

The perimeter of a triangle is $37$ cm. One side is $(2x + 3)$ cm, a second side is $(3x - 1)$ cm, and the third side is $10$ cm. Find $x$ and the length of each side.

Explain why $5(x + 3)$ and $5x + 15$ are equal. What must 15 equal? Now expand $4(x - 2)$ and simplify $4(x - 2) + 6x$.

A number machine applies the rule "$\times 3$, then $- 8$". An input gives an output of $7$. Find the input. Then find the input that gives an output equal to the original input.