National 5 Mathematics: relationships
This set covers the part of National 5 Maths where algebra meets the coordinate grid: solving equations and inequations, handling two equations at once, finding the gradient and the equation of a straight line, and reading a quadratic for its roots, its turning point and how many times it meets the x-axis.
Ten single-answer questions work through exactly those. Two of them are the ones that cost marks most reliably: an inequation divided by a negative, where the sign has to be reversed, and a discriminant, where using the wrong sign inside the formula turns "no real roots" into "two real roots". The simultaneous-equations question is built so that each wrong pair satisfies one of the two equations but fails the other — guessing from a single line does not work. Explanations show the working line by line.
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- Solve a linear equation with a bracket on one side
- Solve an inequation and reverse the sign when dividing by a negative number
- Solve a pair of simultaneous equations by elimination and check both equations
- Find a gradient from two points, including negative coordinates
- Find the equation of a line from its gradient and a point on it
- Read the roots of a factorised quadratic with the correct signs
- State the turning point from vertex form and say whether it is a maximum or a minimum
- Use the discriminant to decide how many real roots an equation has
- Evaluate a function at a negative value
- Solve a quadratic by factorising when the leading coefficient is not 1
How many real roots does the quadratic equation $2x^2 - 5x + 4 = 0$ have? — The discriminant $D = b^2 - 4ac = (-5)^2 - 4(2)(4) = 25 - 32 = -7$. Since $D < 0$, there are no real roots.
Sample question
What is the gradient of the straight line passing through the points $(-1, 4)$ and $(3, -4)$?
- $2$
- $-\frac{1}{2}$
- $\frac{1}{2}$
- $-2$
See the answer
$-2$
The gradient $m$ is given by $\frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 4}{3 - (-1)} = \frac{-8}{4} = -2$.
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↑ National 5 and Higher (Scotland)
- National 5 Maths: Trig Identities, the Sine Rule for an Angle and Bearings
- National 5 Maths: Trigonometric Graphs, Related Angles and Equations
- National 5 Maths: Circle and Shape Geometry, 3D Pythagoras and Volume
- National 5 Maths: Quadratic Formula, Completing the Square and Sketching
- National 5 Maths: Equations and Inequations in Context
- National 5 Mathematics: geometry and trigonometry
National 5 and Higher qualifications are awarded by Qualifications Scotland, which took over the awarding functions of the Scottish Qualifications Authority (SQA) in February 2026. Zestly is not affiliated with or endorsed by Qualifications Scotland or the SQA. These materials are our own and reproduce no past-paper content.