Solving an equation.

Step 1 of 4

Two sides, two lines

An equation says that two expressions are equal: $2x + 1 = 7$⁠. The expression to the left of the equals sign is the left side, and the one to its right is the right side. Solving the equation means finding the $x$⁠ for which the two sides really are equal.

On the board each side is a line: the left side is the blue line $y = 2x + 1$⁠, and the right side is the pink line $y = 7$⁠, a horizontal line, because it has no $x$⁠ in it. At any $x$⁠ you can read off the value of each side. The solution is the $x$⁠ where the two lines meet.

There is also a way without the board: undo what was done to $x$⁠, in reverse order. In $2x + 1$⁠, $x$⁠ was first multiplied by $2$⁠ and then $1$⁠ was added. So first subtract $1$⁠, then divide by $2$⁠. Solve it, then choose Where they meet under Look inside to check.

Step 2 of 4

The same to both sides

An equation is like a pair of scales in balance: you may do anything at all, as long as you do it to both sides. Add $4$⁠ to one side only and the scales no longer balance.

The board now shows $3x - 4 = 5$⁠. To leave $x$⁠ on its own, undo first what was done last: the subtraction of $4$⁠.

Step 3 of 4

x on both sides

Sometimes $x$⁠ appears on both sides, as in $4x - 3 = x + 3$⁠. Now the right side is a sloping line too, and the two lines meet somewhere else.

The method: gather all the $x$⁠ terms on one side and all the numbers on the other, doing the same thing to both sides at every step. Subtract $x$⁠ from both sides: $3x - 3 = 3$⁠. Add $3$⁠ to both sides: $3x = 6$⁠.

Step 4 of 4

Check by substituting

To check a solution, substitute it into both sides: if they come out the same, the solution is right. In $2x + 1 = 7$⁠, substituting $x = 3$⁠ gives $2 \cdot 3 + 1 = 7$⁠, exactly the right side.

You can also turn the question round: you are given the solution and look for the equation. The left side is locked at $2x + 1$⁠, and only Right: number moves.