3.3PureCore

Simultaneous Equations on Graphs

Every simultaneous-equation solution can be seen on a graph: it is exactly where the curves cross. This lesson connects the algebra you have done to the picture, and uses the discriminant to count intersections.

25 min Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.3 Simultaneous Equations on GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Interpret solutions as points of intersection
Count intersections between a line and a curve
Use the discriminant to test for tangency
Find the condition for a line to be a tangent

Solutions are intersection points

Plotting both equations, the simultaneous solutions are the coordinates where the graphs meet. Two graphs can meet at several points — each one is a solution pair.

Counting intersections with the discriminant

Substituting one equation into the other gives a quadratic. Its discriminant $b^{2} - 4 a c$ tells you the number of intersection points without drawing anything.

b^{2} - 4 a c > 0 (2), = 0 (1, tangent), < 0 (0)

The discriminant of the combined quadratic counts the intersections.

1Set equal:

x^{2} = x + k \Rightarrow x^{2} - x - k = 0

2Discriminant:

(- 1)^{2} - 4 (1) (- k) = 1 + 4 k

3Two points if

1 + 4 k > 0

, tangent if

= 0

, none if

< 0

Answertangent when

k = - \frac{1}{4}

Tip — Tangent ⇔ discriminant = 0. This is a favourite exam phrasing.

Finding a tangency condition

If a line is a tangent to a curve, set the combined quadratic’s discriminant to zero and solve for the unknown. This finds the exact value that makes the line just touch.

Formula recap

solutions = intersection points

Algebra meets geometry.

b^{2} - 4 a c = 0 \Leftrightarrow tangent

One repeated intersection.

Common mistakes to avoid

Thinking two curves can only meet once.

A line and a parabola can meet 0, 1 or 2 times — use the discriminant to decide.

Forgetting that “touches” means a repeated root.

Touching/tangent means discriminant = 0, not > 0.

Key takeaways

Solutions of simultaneous equations are the graphs’ intersection points.
Substitute to form a quadratic, then use its discriminant to count intersections.
Discriminant = 0 is the tangent condition.

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