Solving equations and inequalities: solve linear equations in one unknown algebraically find approximate solutions using a graph

The idea

This skill has two linked parts. Algebraically, you solve a linear equation in one unknown by using inverse operations to isolate the variable, keeping the equation balanced at every step. For example, to solve 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Graphically, you can treat each side as a function, plot y = 3x + 5 and y = 20, and read the x-coordinate where the lines cross; that x-value is the approximate solution. The graph method is useful when an exact algebraic solution is difficult or when you need to check your answer. Both methods should give the same value, so you can use one to verify the other.

What earns the marks

• Use inverse operations in the correct order to isolate the unknown, keeping the equation balanced.
• Check the algebraic solution by substituting it back into the original equation.
• For a graphical method, plot each side of the equation as a separate straight-line graph on the same axes.
• Read the x-coordinate of the point of intersection to obtain the approximate solution.
• Recognise that the graphical solution may be approximate because it depends on reading accuracy.

Common mistakes and how to put them right

• Applying an operation to only one side of the equation.

  Put it right by performing the same operation to both sides.

• Dividing before adding or subtracting when the unknown has a coefficient.

  Put it right by undoing addition or subtraction first, then multiplication or division.

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

  Put it right by identifying that the solution is the x-value where the two graphs meet.

• Assuming a graph always gives an exact answer.

  Put it right by stating that graphical solutions are approximate and should be checked algebraically where possible.

Exam tips

• Show each algebraic step clearly so that method marks can be awarded even if the final answer is wrong.
• When using a graph, draw the lines accurately and label the intersection point before reading the x-value.
• Substitute your solution back into the original equation to confirm it works, especially in multiple-choice questions where options may be close.