Lines that never meet
So far every pair of equations has had exactly one solution, one point where the two lines cross. That is not always so.
Take 2x + y = 5 and 4x + 2y = 14. Draw both lines and they are equally steep: they run side by side, the same distance apart, and never meet. Lines like this are parallel. With no crossing point, no pair of values makes both equations true.
The gold line is 2x + y = 5. The two lines are equally steep, so they never cross.
Scale one equation to see it
You do not need to draw the lines to find this out. Double every term of the first equation: 4x + 2y = 10. Now its letter terms, 4x + 2y, are exactly the letter terms of the second equation.
So the same amount, 4x + 2y, would have to equal 10 and also equal 14. That is impossible, so the pair of equations has no solution. Trying to eliminate a letter shows the same thing: subtracting 4x + 2y = 10 from 4x + 2y = 14 removes both letters at once and leaves 0 = 4, which is false.
One line drawn twice
Now change the second equation to 4x + 2y = 10. Doubling the first equation gives 4x + 2y = 10, which is the second equation exactly. The two equations say the same thing in different sizes, so they draw one single line.
Every point on that line is on both lines, so every pair of values that solves one equation solves the other. For example, x = 0 and y = 5, x = 1 and y = 3, and x = 2 and y = 1 all work. The pair of equations has infinitely many solutions. Subtracting them leaves 0 = 0, which is true whatever x and y are.
4x + 2y = 10 is drawn exactly on top of the gold line, 2x + y = 5: the two equations draw one and the same line.
Exactly one crossing
Now take 2x + y = 5 and 3x + y = 8. The y terms already match, but no number multiplies 2x to give 3x while leaving y as it is. The letter terms can never be made the same, so the lines are not equally steep, and they cross exactly once.
Subtract the first equation from the second to find the crossing: x = 3. Then 2 × 3 + y = 5, so y = −1. The one solution is x = 3 and y = −1.
The gold line is 2x + y = 5 again. Lines with different steepness cross once, here where x = 3 and y = −1.
Three cases, one test
Try to scale one equation so that its letter terms match the other's. If they can be matched and the numbers on the right match too, the equations are the same line, with infinitely many solutions. If the letter terms match but the numbers do not, the lines are parallel, with no solution. If the letter terms cannot be matched, the lines cross once, and there is exactly one solution.
m₁ ≠ m₂: the two lines share exactly one point, and its coordinates are the one pair (x, y) that satisfies both equations
Make the gradients equal with different intercepts
The line y = x + 1 stays fixed. One handle turns the gold line, and the other slides it up and down. Turn it until it is exactly as steep as y = x + 1, and the lines no longer meet. Slide it onto y = x + 1, and every point is shared.
The usual mistakes
Reading 0 = 4 as a slip in the working and starting again. When every letter cancels and a false statement is left, the working is correct: it shows that the equations have no solution.
Deciding from how the equations look. 2x + y = 5 and 4x + 2y = 10 look different, but halving every term of the second gives the first, so they have infinitely many solutions, not one and not none.
Worked example: Two Data Plans with the Same Rate That Never Cost the Same
Question Plan A for mobile data costs $18 a month and then $2 for each gigabyte used. Plan B costs $25 a month and then $2 for each gigabyte used. (a) Show that there is no amount of data for which the two plans cost the same, and find how much more Plan B always costs. (b) A leaflet describes Plan C by the equation 2y − 4x = 36, where y dollars is the monthly cost for x gigabytes. How many pairs of values (x, y) satisfy the equations of both Plan A and Plan C?
1.Let x be the number of gigabytes used and y the monthly cost in dollars. Plan A is y = 2x + 18 and Plan B is y = 2x + 25.
For x gigabytes, Plan A costs y = 2x + 18 dollars and Plan B costs y = 2x + 25 dollars. 2.If the two plans cost the same, then 2x + 18 = 2x + 25. Subtract 2x from both sides: 18 = 25. This statement is false, whatever the value of x.
Setting the costs equal and subtracting 2x from both sides leaves 18 = 25, which is false. 3.(a) No value of x satisfies both equations, so the pair of equations has no solution. Both lines have gradient 2, and their intercepts, 18 and 25, are different, so the lines are parallel. Plan B always costs 25 − 18 = $7 more.
(a) There is no solution. The lines have the same gradient, 2, and different intercepts, so they are parallel, and Plan B always costs $7 more. 4.For Plan C, divide both sides of 2y − 4x = 36 by 2: y − 2x = 18. Add 2x to both sides: y = 2x + 18. This is exactly the equation of Plan A.
Divide 2y − 4x = 36 by 2 and add 2x to both sides: y = 2x + 18, the equation of Plan A. 5.(b) The two equations describe the same line, so every point on it satisfies both: there are infinitely many solutions. Setting the costs equal gives 18 = 18, which is true for every value of x.
(b) Every point on the line satisfies both equations, so there are infinitely many solutions.
Answer: (a) no solution: the lines are parallel, and Plan B always costs $7 more; (b) infinitely many
Common mistakes
- Reading 18 = 25 as a slip in the algebra and starting again. A false statement with no x left in it is the algebra showing that the equations have no solution.
- Saying that Plan A and Plan C have no solution because 2y − 4x = 36 looks different from y = 2x + 18. Dividing every term by 2 shows that they are the same equation, so there are infinitely many solutions.
More equations and inequalities problems, worked step by step →