Solving equations and inequalities: including those with the unknown on both sides of the equation

The idea

When the unknown appears on both sides of a linear equation, first collect the unknown terms on one side and the constant terms on the other. For example, to solve 5x + 2 = 3x + 10, subtract 3x from both sides to get 2x + 2 = 10, then subtract 2 from both sides to get 2x = 8, and finally divide by 2 to get x = 4. The same balance principle applies: whatever operation you perform on one side must be performed on the other. You can also solve such equations graphically by plotting each side as a straight line and reading the x-coordinate of their intersection. Always check your solution by substituting it back into the original equation.

What earns the marks

• Collect the unknown terms on one side of the equation by adding or subtracting the same term from both sides.
• Collect the constant terms on the other side by adding or subtracting the same number from both sides.
• Divide or multiply both sides by the coefficient of the unknown to isolate it.
• Check the solution by substituting it back into the original equation and confirming both sides are equal.
• For a graphical approach, plot each side as a straight line and read the x-coordinate of the intersection point.

Common mistakes and how to put them right

• Moving a term without changing its sign.

  Put it right by remembering that adding or subtracting a term from both sides reverses its sign on the side it is moved from.

• Collecting unknowns on the side that makes the coefficient negative.

  Put it right by choosing to collect on the side that keeps the coefficient positive where possible, or by carefully handling negative coefficients.

• Forgetting to divide the constant term by the coefficient.

  Put it right by dividing every term by the coefficient of the unknown.

• Reading the y-coordinate of the intersection point on a graph instead of the x-coordinate.

  Put it right by identifying that the solution is the x-value where the two lines cross.

Exam tips

• Write down each step of your rearrangement so that method marks can be awarded even if the final answer is incorrect.
• After solving, substitute your value back into the original equation to check both sides are equal.
• In multiple-choice questions, substitute each option into the equation to see which one works, especially if you are unsure of your algebraic method.