Simultaneous linear equations

• Elimination: make the coefficients of one letter match, then add or subtract the equations.
• Substitution: rearrange one equation for a letter and substitute it into the other.
• Always substitute back to find the second value, and check in the other equation.
• Higher: the solution is the point of intersection of the two straight lines.

Exact solutions

Elimination: if the coefficients of one letter are equal, subtract the equations; if they are equal but opposite in sign, add them ("Same Signs Subtract"). If they don't match, multiply one or both equations first. Substitution is quicker when one equation already gives a letter on its own or with coefficient 1.

Multiply both equations first

When neither letter has matching coefficients, multiply each equation by a number to make one pair match — like finding a common denominator. Number the equations so your working is easy to follow.

Where two lines meet

Each linear equation is a straight line. The pair (x, y) that satisfies both equations lies on both lines, so it is their point of intersection. Parallel lines (same gradient, different intercept) never meet, so their equations have no solution.