Mathematics · Lesson 10

Simultaneous equations, inequalities & linear programming

JAMBWAECNECO

What the syllabus asks

JAMB: simultaneous equations, inequalities, linear programming

WAEC: Algebraic processes: simultaneous linear equations in two variables (elimination, substitution, graphical), linear and quadratic simultaneous equations, linear inequalities in one and two variables, graphical representation, linear programming

NECO: Algebraic processes: simultaneous linear equations in two variables (elimination, substitution, graphical), linear and quadratic simultaneous equations, linear inequalities in one and two variables, graphical representation, linear programming

What you will learn

Solve two linear equations by elimination, substitution and graph.
Form and solve simultaneous equations from word problems.
Solve one linear and one quadratic equation simultaneously.
Solve linear inequalities and show solutions on a number line.
Represent inequalities in two variables as regions.
Solve linear programming problems by testing vertices.

Key points

Simultaneous equations must all be satisfied at once; graphically, their solution is the intersection.
Elimination: equalise one coefficient, then add (opposite signs) or subtract (same signs).
Substitution: make one variable the subject and substitute; best with linear–quadratic pairs.
Clear fractions first; for reciprocals let a = 1/x, b = 1/y.
In word problems, define letters and write one equation per fact.
Inequalities: reverse the sign when multiplying or dividing by a negative.
Regions: solid line for ≤/≥, broken for ; test a point such as (0, 0).
Linear programming: optimum of the objective function is at a vertex of the feasible region.

Start here: What simultaneous equations are

One equation with two unknowns, such as x + y = 10, has endless solutions: (1, 9), (2, 8), (3.5, 6.5) and so on. If we add a second fact, for example x − y = 4, only one pair satisfies both: x = 7, y = 3. Equations that must be true at the same time are called simultaneous equations.

Think of a market problem. Mama Bisi sells oranges and pineapples. You know two things: 3 oranges and 1 pineapple cost ₦900, and 1 orange and 1 pineapple cost ₦700. Neither fact alone tells you the prices, but together they do: subtracting the second from the first removes the pineapple, so 2 oranges cost ₦200, and one orange is ₦100. Then a pineapple is ₦600. That subtraction is exactly the elimination method.

Graphically, each linear equation is a straight line. The solution is the point where the two lines cross, because only that point lies on both lines.

Two lines can: cross once (one solution), be parallel (no solution), or be the same line (infinitely many solutions).

Worked example 10.1

Check whether (3, 2) solves 2x + 3y = 12 and 3x − y = 7.

2(3) + 3(2) = 6 + 6 = 12 ✓.
3(3) − 2 = 9 − 2 = 7 ✓.

Yes, (3, 2) satisfies both

Worked example 10.2

How many solutions do x + 2y = 5 and 2x + 4y = 7 have?

Double the first: 2x + 4y = 10.
But the second says 2x + 4y = 7. Both cannot be true: the lines are parallel.

No solution

Try it yourself. Does (1, 4) satisfy x + y = 5 and 2x − y = 1?

Show solution

1 + 4 = 5 ✓ but 2 − 4 = −2 ≠ 1. No.

Continue this lesson in the free app →

Practice questions

1. Solve 3x + 2y = 13 and x − y = 1.

Show worked answer

x = 3, y = 2.

2. Solve 2x + 5y = 26 and 4x − 3y = 0.

Show worked answer

x = 3, y = 4.

3. Solve y = x − 2 and x² + y² = 20.

Show worked answer

x² + (x − 2)² = 20 → x² − 2x − 8 = 0: (4, 2) and (−2, −4).

4. Solve 5 − 2x ≤ 11 − 4x.

Show worked answer

2x ≤ 6: x ≤ 3.

5. List the integers satisfying −1 ≤ 3x + 2 < 11.

Show worked answer

−1 ≤ x < 3: −1, 0, 1, 2.

Key words

Simultaneous equations: Two or more equations solved together for common values.
Elimination: Removing one variable by adding or subtracting equations.
Substitution: Replacing a variable by an equivalent expression from another equation.
Consistent: Having at least one solution.
Inequality: A statement using <, >, ≤ or ≥.
Critical value: A boundary value where an inequality changes truth.
Feasible region: The set of points satisfying all constraints.
Constraint: A condition (inequality) that limits the variables.
Objective function: The quantity to be maximised or minimised, e.g. profit P = 3x + 2y.
Vertex: A corner point of the feasible region.
Linear programming: A method of finding the best value of a linear objective under linear constraints.
Boundary line: The line ax + by = c that bounds a region.

Learn Simultaneous equations, inequalities & linear programming step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android