Simultaneous Equations

Simultaneous Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A simultaneous equation is a pair of linear equations in the same variables whose common solution satisfies both at once. For two variables, the standard form is:

ax + by = c and dx + ey = f

where a, b, c, d, e, f are known constants and (x, y) is the unknown pair. Two algebraic routes dominate NABTEB: the substitution method (solve one equation for one variable, then plug into the other) and the elimination method (multiply equations by constants so adding or subtracting cancels one variable). When the determinant D = ae − bd ≠ 0, the system is consistent with a unique solution given by x = D_x/D and y = D_y/D, where D_x = ce − bf and D_y = af − cd (this is Cramer’s rule). Quick exam pointers: (1) always check D first; (2) when D = 0, there is either no solution (parallel lines) or infinitely many (coincident lines); (3) word problems on ages, costs, and mixtures translate to this form and usually carry 3–5 marks.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Form and System Types

A pair of linear equations in two unknowns is written:

ax + by = c … (1) dx + ey = f … (2)

The coefficient matrix has determinant D = ae − bd. Three cases arise:

Value of D	D_x, D_y	Nature	Graphical Picture
D ≠ 0	any	Consistent, unique solution	Two lines intersect at one point
D = 0	both zero	Consistent, infinitely many solutions	Coincident lines (same line)
D = 0	at least one non-zero	Inconsistent, no solution	Parallel distinct lines

Substitution Method

From equation (1), express y = (c − ax)/b (provided b ≠ 0). Substitute into (2) to obtain a single equation in x. Solve for x, then back-substitute to find y. This method is fastest when one variable has coefficient 1, e.g. y = 3x − 2.

Elimination Method

Multiply (1) and (2) by chosen constants so that one variable’s coefficients become equal in magnitude. Adding eliminates that variable when the signs are opposite; subtracting eliminates it when the signs are the same. Solve the resulting single-variable equation, then substitute back.

Cramer’s Rule

Compute the three determinants:

D = ae − bd, D_x = ce − bf, D_y = af − cd

Then x = D_x / D and y = D_y / D. This is the quickest paper-friendly method for multiple-choice and short-answer NABTEB items.

Worked Example

Solve: 3x + 2y = 12 and x − y = 1. Using elimination, multiply the second equation by 2: 2x − 2y = 2. Add to the first: 5x = 14, so x = 14/5. Substituting back: 14/5 − y = 1 ⇒ y = 9/5. Check: 3(14/5) + 2(9/5) = 42/5 + 18/5 = 60/5 = 12 ✓.

Exam Question Patterns

NABTEB tests this topic through (i) direct “solve for x and y” items, (ii) word problems on ages, costs of items, mixtures, and rates, and (iii) occasional graphical interpretation questions. Each item typically carries 2–6 marks; word problems are usually 4–5 marks.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Determinant Behaviour

When the system reduces to 0x + 0y = k (k ≠ 0) after elimination, the system is inconsistent — no (x, y) pair can satisfy it. This corresponds to two parallel lines on the Cartesian plane that never meet. When the elimination yields 0x + 0y = 0, the second equation is a scalar multiple of the first, giving infinitely many solutions along a single line. Students commonly mis-apply Cramer’s rule here and divide by zero, which is undefined; the correct response is to state “no unique solution” and classify the system.

Connections to Adjacent Topics

Simultaneous equations connect directly to coordinate geometry (the slope-intercept form y = mx + c), matrix algebra (the coefficient matrix [[a, b], [d, e]]), and linear inequalities (the boundary lines of a feasible region are simultaneous-equation lines). The inverse of a 2×2 matrix equals (1/D) × [[e, −b], [−d, a]], which is precisely the algebraic engine behind Cramer’s rule.

Common Mistakes

1. Sign errors when subtracting equations after multiplication — always wrap the substituted equation in brackets before expanding.
2. Incomplete scaling — multiplying only some terms on one side of the equation.
3. Mixing D_x and D_y — D_x replaces the x-coefficients (a, d) with constants (c, f), while D_y replaces the y-coefficients (b, e) with constants.
4. Word-problem mistranslation — confusing “twice as old” with “two years older” or mixing per-unit costs with totals.

Practice Prompts

1. A shop sells 3 pens and 2 pencils for ₦230, and 2 pens and 4 pencils for ₦260. Find the cost of one pen and one pencil.
2. Solve the system 4x − 3y = 10 and 2x + 5y = −6 using Cramer’s rule, and verify your answer by substitution.

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