Algebra
Pair of linear equations
Solve a pair of linear equations in two variables by substitution and elimination, and say how many solutions a pair has.
One equation with two letters in it has endless solutions: x + y = 10 is true for 1 and 9, for 4 and 6, for 7.5 and 2.5. Draw them all and you get a line. A second equation gives a second line, and now the question is a sharp one — which single pair of numbers satisfies both at once? That is where the two lines cross.
Two ways in
Substitution: rearrange one equation to get a letter on its own, then put that into the other. Elimination: add or subtract the equations so that one letter cancels. Same answer, and elimination is usually quicker when the coefficients already match.
Have a play
One scale, and every box on it hides the same weight. Move things about and find out what a single scale can and cannot tell you.
Worked example
Solve 2x + y = 11 and x - y = 1.
Add the two equations: (2x + y) + (x - y) = 11 + 1, so 3x = 12.
The +y and the -y cancel each other. That is the whole trick of elimination — you choose which letter to lose.
Not every pair crosses. Two lines can be parallel, in which case there is no solution at all, or they can be the same line drawn twice, in which case every point on it is a solution. You can tell which before solving anything, just by comparing the coefficients.
How to tell which
Write the pair as a₁x + b₁y = c₁ and a₂x + b₂y = c₂. If a₁/a₂ and b₁/b₂ differ, the lines cross once. If all three of a₁/a₂, b₁/b₂ and c₁/c₂ agree, it is one line twice and there are infinitely many solutions. If the first two agree but c₁/c₂ does not, the lines are parallel and there is no solution.
Try it together
The sum of two numbers is 25 and their difference is 7. Let us find them.
Call the larger x and the smaller y, and turn each sentence into an equation.
1.Adding the two equations gives 2x on the left. What is 25 + 7?
Have a go
Have a go on your own: solve the pair 4x + y = 14 and 2x - y = 4. What is x?
Hint: Add the equations and the y terms cancel.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.