6 mistakes learners make with solving linear equations, each one named and explained.
An equation is a balance: whatever you do to one side you do to the other, and you undo the operations in the reverse order to how they were applied.
5x - 4 = 2x + 11 is answered as x = 1, because the terms were combined rather than moved.
Both sides have x terms and moving one across a sign is the step where the sign is most easily dropped.
Question. 5x - 4 = 2x + 11
A common answer. 2x taken from the left only, giving 3x - 4 = 2x + 11
The answer. 3x = 15, so x = 5
Why. Taking 2x from one side means taking it from both, which is what keeps the balance true.
Whatever you take from one side comes off the other. That is the whole rule.
4x + 7 = 31 is solved by adding 7, or a number multiplied by 4 is undone by multiplying by 4 again.
The operation in the question is the one in front of you, and undoing means doing something, so you do that.
Question. 4x + 7 = 31
A common answer. adding 7 to give 4x = 38
The answer. subtracting 7 to give 4x = 24
Why. The 7 was added on, so taking it off is what leaves 4x on its own.
Every step is undone by its opposite. Always check by putting your answer back in.
3x + 7 = 22 is answered as x = 15, which is 22 - 7 with the dividing never done.
The hard step was the subtraction and it worked, so it feels like the end of the job.
Question. 3x + 7 = 22
A common answer. x = 15
The answer. x = 5
Why. 22 - 7 leaves 3x = 15, and x is a third of that.
The answer is x on its own. If there is still a number stuck to it, you are not finished.
The two equations are combined the wrong way, so neither unknown is eliminated.
Elimination sometimes adds and sometimes subtracts, and only the signs of the y terms decide.
Question. 2x + y = 11 and x - y = 1
A common answer. subtracting, giving x + 2y = 10
The answer. adding, giving 3x = 12
Why. The y terms are plus y and minus y, so adding cancels them.
Look at the signs of the terms you want to remove. They decide the operation, nothing else does.
y is given where x was asked for, or the pair is reported the wrong way round.
Both values come out of the same working and both are correct answers to something.
Question. 2x + y = 11 and x - y = 1, find x
A common answer. 3, which is y
The answer. 4
Why. Adding the equations gives 3x = 12, so x is 4 and y is 3.
Label each value as you find it. The question names which one it wants.
A number is multiplied by 4 then 7 is added to give 39, and you work out (39 + 7) divided by 4.
The operations are undone in the order they are read rather than in reverse, which is a subtle thing nobody says out loud.
Question. multiplied by 4, then 7 added, giving 39
A common answer. (39 + 7) / 4
The answer. (39 - 7) / 4
Why. The adding happened last, so it is undone first.
Unwrap in reverse, like taking off shoes and socks in the order they went on.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.