Solving equations and inequalities: solve two simultaneous equations in two variables (linear/linear) algebraically find approximate solutions using a graph

The idea

This topic requires you to solve a pair of linear simultaneous equations in two variables algebraically, and to use a graph to find approximate solutions. Algebraically, use elimination or substitution. For example, to solve 2x + y = 7 and x − y = 2, adding the equations eliminates y, giving 3x = 9, so x = 3; substituting back gives y = 1. Graphically, draw both straight lines on the same axes; the point where they cross gives the approximate solution. You should be able to read off coordinates from a graph, recognising that the intersection may not be exact. The algebraic method gives an exact answer, while the graph gives an approximate answer. Always check your solution by substituting back into both original equations.

What earns the marks

• Correctly eliminate one variable by adding or subtracting the equations, or by substitution, to form a single equation in one variable.
• Solve the resulting linear equation accurately to find the value of one variable.
• Substitute the found value back into one of the original equations to find the other variable.
• State the solution as a coordinate pair (x, y) and verify by substitution into both original equations.
• For graphical solutions, plot both lines accurately and read the coordinates of the intersection point, giving an approximate answer.

Common mistakes and how to put them right

• Forgetting to multiply both sides of an equation when using elimination, leading to an incorrect elimination step.

  Put it right: Apply the same operation to the entire equation.

• Substituting the found value into the wrong equation or making an arithmetic error.

  Put it right: Substitute into the simpler equation and check in both.

• Reading the intersection point from a graph inaccurately, for example misreading the scale.

  Put it right: Use a ruler and check that the point lies on both lines.

• Giving the solution as two separate numbers without indicating which is x and which is y.

  Put it right: Write the answer as a coordinate pair (x, y).

Exam tips

• Show your working clearly, especially the elimination or substitution steps, as method marks may be available.
• When using a graph, draw the lines with a ruler and label them; read the intersection carefully and state that it is approximate.
• Always check your solution by substituting back into both original equations; this can catch arithmetic errors.