When should I use elimination vs substitution?

Use elimination for two linear equations (multiply and add/subtract to cancel a variable). Use substitution when one equation is already solved for a variable, or one is linear and the other quadratic. For linear-quadratic systems, substitution into the linear equation is mandatory to avoid the plug-back trap.

What does the discriminant tell me about simultaneous equations?

After substituting the linear equation into the quadratic, you get a quadratic in one variable. The discriminant Δ = b² − 4ac tells you how many real solutions exist: Δ > 0 means 2 intersections, Δ = 0 means the line is tangent (1 intersection), Δ < 0 means no intersection. This is frequently tested in Paper 2 and Section B.

Why must I check the domain after solving log or fractional equations?

Logarithms require strictly positive arguments (> 0); fractional denominators cannot be zero. After solving, substitute every solution back into the original expression. Reject any root that violates these conditions — this is called rejecting extraneous roots and failing to do so costs the final accuracy mark in Paper 1.

More about Equations

Q: Why must I check the domain after solving log or fractional equations?

Logarithms require strictly positive arguments (> 0); fractional denominators cannot be zero. After solving, substitute every solution back into the original expression. Reject any root that violates these conditions — this is called rejecting extraneous roots and failing to do so costs the final accuracy mark in Paper 1.

Master linear-quadratic simultaneous equations, unmask hidden quadratic equations in disguise, and learn how to systematically check your roots to avoid losing easy marks.

1. The Core Concept

The Power of Substitution: Solving complex equations is all about unmasking the underlying structure. When solving a linear-quadratic system, we substitute the linear "clue" to reduce two variables into a single quadratic equation. When solving equations in disguise, we use the "Let u = …" substitution trick to transform fractional, exponential, logarithmic, or trigonometric terms into a standard quadratic form (au² + bu + c = 0).

2. DSE More about Equations Cheat Sheet — 5 Key Topics

Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.

Topic	Key Concept & Steps	Exam Shortcuts & Tips
1. Linear-Quadratic	Make `y` (or `x`) the subject of the linear equation and substitute it into the quadratic equation: `ax² + bx + c = 0`	Intersection of a straight line and a curve (parabola or circle). Do not plug x back into the quadratic: Always substitute solved `x` values back into the linear equation to find `y` to avoid pairing errors.
2. Fractional	Multiply the entire equation by the LCM of the denominators to clear the fractions.	Analyzing rational curves. Always check your denominators: Any solution that makes a denominator equal to 0 must be rejected.
3. Exponential	Let `u = aˣ` to transform: `p(a²ˣ) + q(aˣ) + r = 0 ⟹ pu² + qu + r = 0`	Analyzing rapid growth curves. Reject negative u values: Since `aˣ > 0` for all real `x`, any negative solution (e.g., `2ˣ = −1`) has no real solution.
4. Logarithmic	Condense log terms using laws first: `log M + log N = log(MN)` If `log M = log N ⟹ M = N`	Analyzing logarithmic curves. Verify the domain strictly: The argument of any log must be strictly positive. Reject any roots that make an argument ≤ 0.
5. Trigonometric	Let `u = sin θ` (or `cos θ`) to transform: `pu² + qu + r = 0`	Analyzing repeating wave curves. Reject u outside [−1, 1]: Since `−1 ≤ sin θ ≤ 1` and `−1 ≤ cos θ ≤ 1`, reject any `u` value outside this range, then solve for `θ`.

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💡 Exam Traps & Secrets

⚠️ The Linear-Quadratic Pairing TrapWhen you solve a linear-quadratic system, you will obtain two values for x (e.g., x = 2 or x = −1). You must plug these values back into the linear equation to find the corresponding y values.
The Trap: If you substitute them into the quadratic equation, you will generate extra, incorrect coordinate pairs and lose the final accuracy marks.

⚠️ The Logarithmic Domain CheckWhen you solve a logarithmic equation, you must check your final solutions against the original log arguments.
Example: If you solve log(x − 2) + log(x + 1) = log 10 and obtain x = 4 or x = −3:
At x = 4 ⟹ log(4 − 2) = log(2) (valid).
At x = −3 ⟹ log(−3 − 2) = log(−5) (invalid!). You must reject the extraneous root x = −3.

⚠️ The "Does Not Exist" / Tangency ConditionIf a question asks for the condition where a line and a curve touch at exactly one point (or are tangent), substitute to form a quadratic and set the discriminant to zero:
Δ = b² − 4ac = 0

⚠️ Word Problem Boundary ChecksWhen translating real-world problems (e.g., rectangle dimensions) into quadratic equations, check if your answers make physical sense. Dimensions, lengths, times, or counts cannot be negative. Always reject negative solutions and write a clear concluding sentence.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Linear-Quadratic substitution & coordinate pairing)Solve the simultaneous equations: y − x = 1 and y = x² − 1.

(Quadratic substitution with u = 2ˣ, rejecting negative roots)Solve the exponential equation 4ˣ − 3 · 2ˣ − 4 = 0.

(Condensing log terms, quadratic factorization, rejecting extraneous roots)Solve the logarithmic equation log(x − 2) + log(x + 1) = log 10.

(Δ = 0)Find the value(s) of k for which the line y = kx + 3 and the parabola y = x² − 2 have exactly one solution.

Frequently Asked Questions

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