This question asks you to place a point on a number line. The number is not an integer.

Taken from question 37 of the book Problem Solving in GCSE Mathematics by Daniel Griller.

Given bearings and lengths of two straight lines, work out the bearing and distance back to the starting point.

A Eukleides diagram shows the setup visually.

Solve linear equations with one unknown. Including brackets and fractions.

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Given 3 vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$, find the constants $p$, $q$ and $k$ such that $ p\mathbf a + q \mathbf b = \mathbf c$, where $k$ is an unknown component of $\mathbf c$ .

Given 3 vectors $\mathbf v$, $\mathbf a$ and $\mathbf b$, find the constants $c_1$ and $c_2$ such that $\mathbf v = c_1 \mathbf a + c_2 \mathbf b$ .

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Manipulation of algebraic fractions

(a) Equation given, and five coordinates. Student should select those coordinates which lie on the line. (b) Gradient and a point on line is given. Student is to calculate coordinates of other points on the line.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

This question uses a Geogebra applet to solve a linear program with two variables using the graphical method. It contains three steps:

1. Construct the feasible area (polygon) by adding the constraints one by one. The students can see what happens when the constraints are added.
2. Add the objective function, and the level set of the objective value is shown, as well as its (normalised) gradient.
3. Compute the optimal solution by moving the level set of the objective around.

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Solve linear equations with unkowns on both sides. Including brackets and fractions.

Solving a pair of simultaneous equations of the form $a_1x+y=c_1$ and $a_2x^2+b_2xy=c_2$ by forming a quadratic equation.

Solving a pair of linear simultaneous equations, giving answers as integers or fractions.

Solve linear equations with unkowns on both sides. Including brackets and fractions.

Solve linear equations with unkowns on both sides. Including brackets and fractions.

Finding composite functions of 2 linear functions.

Evaluating a linear function for a given value of $x$.

Calculating gradient and finding intercept from a geogebra graph.

Expanding two linear brackets multiplied together.

Classic problem of a vehicle parked on an incline.  Best solved by rotating the coordinate system.

Image Credit: https://svgsilh.com/image/34325.html  CC-0

Simple substitution into an algebraic expression. Includes powers, division, mulitplication, brackets.  Includes subscripts.

You can enter an equation in 2 or more variables and rearrange it in the following steps. Numbas checks that the lines of working are all equivalent to the original equation. 

Solving simple linear equations in $\mathbb{Q}$ and $\mathbb{Z}_n$ for $n= 13, \;17$ or $19$.