B: Algebra and functions: Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

The idea

This topic requires you to solve pairs of equations in two variables. For two linear equations, elimination is often efficient: multiply one or both equations to make the coefficients of one variable equal, then add or subtract to eliminate it. Substitution is also valid: rearrange one equation to make a variable the subject and substitute into the other. When one equation is linear and the other is quadratic, substitution is usually the best method. Rearrange the linear equation to express one variable in terms of the other, substitute into the quadratic equation, and solve the resulting quadratic. Each solution pair must satisfy both original equations.

For example: Solving y = x + 1 and x² + y² = 25 gives x² + (x+1)² = 25, leading to 2x² + 2x − 24 = 0, so x = 3 or x = −4, with corresponding y values 4 and −3.

What earns the marks

• Solve two linear simultaneous equations by elimination, correctly scaling equations and adding or subtracting to eliminate one variable.
• Solve two linear simultaneous equations by substitution, rearranging one equation and substituting into the other.
• Solve a pair of simultaneous equations where one is linear and one is quadratic by substituting the linear expression into the quadratic equation.
• Form and solve the resulting quadratic equation correctly, including expanding brackets and collecting like terms.
• Find both variables for each solution and present the solutions as coordinate pairs, checking that each pair satisfies both original equations.
• Interpret the number of solutions geometrically, for example a line intersecting a circle or parabola at two points, one point or no points.

Common mistakes and how to put them right

• When eliminating, forgetting to multiply the constant term as well as the variable terms.

  Put it right: Multiply the entire equation by the same factor.

• Substituting into the same equation instead of the other equation.

  Put it right: Substitute the rearranged expression into the other equation to avoid a trivial identity.

• Losing solutions when solving the resulting quadratic, for example by dividing by a variable that could be zero.

  Put it right: Factorise or use the quadratic formula, and consider all cases.

• Not pairing x and y values correctly after solving for one variable.

  Put it right: Substitute each value back into the linear equation to find the corresponding value of the other variable.

Exam tips

• For one linear and one quadratic equation, always use substitution; elimination is usually not suitable.
• Show your substitution step clearly, including the expansion and simplification, to gain method marks even if the final answer is wrong.
• Check each solution pair in both original equations; this helps you spot arithmetic errors and confirms the number of solutions.
• If the resulting quadratic has no real roots, state that there are no real solutions and relate this to the graphs not intersecting.