This course is about equations with two unknowns. It begins with a single linear equation in two variables, its infinitely many solutions and its straight-line graph. It then pairs two equations, explains intersecting, parallel and coincident lines, and shows how the ratios of the coefficients tell you which case you have. Three lessons teach the three algebraic methods, substitution, elimination and cross multiplication, solving the same pairs each way so you can see they agree. The last lesson clears fractions and handles equations with the variables in the denominator by a change of variable. Worked examples and practice with answers throughout.
Middle Years
Solving Linear Equations in Two Variables
Linear equations in two variables and their straight-line graphs, how to tell whether a pair has one, none or infinitely many solutions, and the substitution, elimination and cross multiplication methods.
- 6 lessons
Overview
Lessons
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Linear equations in two variables
The form ax + by + c = 0, solutions as ordered pairs, why there are infinitely many, tables of values, and why the graph is always a straight line.
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Pairs of equations and their solutions
A pair of linear equations, its solution as the meeting point of two lines, and the ratio tests for intersecting, parallel and coincident lines.
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The substitution method
Write one variable in terms of the other, substitute it in the other equation, solve, and substitute back. Worked on 3x + 2y = 5, 2x − 3y = 7 and more.
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The elimination method
Multiply the equations to make one coefficient match, then add, or change the signs of the lower equation and add, so that one variable disappears.
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The cross multiplication method
Write the coefficients in the order b, c, a, b, cross multiply neighbouring columns, and read off x and y from one formula. When the method breaks down and why.
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Equations reducible to linear form
Clear fractional coefficients, and solve pairs with x and y in the denominator by letting 1/x = p and 1/y = q, including brackets and square roots.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Solving Linear Equations in Two Variables
Typed from the teacher's handwritten class slides, 10 pages.
Download the notes (PDF, 357.3 KB)