Most tested M4.15

Solving Simultaneous Equations

This topic covers translating real-world scenarios into algebraic equations and solving them. It focuses on finding the specific values for unknown variables that satisfy one or more equations simultaneously, using both algebraic manipulation and graphical interpretation.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • The first step is often converting a word problem into a mathematical equation or system of equations. Identify the unknowns and assign them variables (e.g., x, y).
  • Linear simultaneous equations (two lines) can be solved algebraically by either 'substitution' (rearranging one equation to make a variable the subject and substituting it into the other) or 'elimination' (adding or subtracting the equations to cancel out one variable).
  • For a system with one linear and one quadratic equation, substitution is the standard method. Rearrange the linear equation and substitute it into the quadratic one. This will typically result in a new quadratic equation to solve.
  • Graphically, the solution(s) to a system of equations are the coordinates of the intersection points of their graphs. A line and a parabola, for instance, can intersect at 0, 1, or 2 points, corresponding to 0, 1, or 2 real solution pairs.
  • Always present your final solutions as pairs of values, for example,. A linear-quadratic system often has two such pairs.
    x=a, y=b
  • After finding a solution, substitute the values back into the original equations to verify they are correct.

Diagram

GraphGraph with axes x and y. solutionxy
Two lines intersect at the solution point. The x and y coordinates of this point represent the values that satisfy both simultaneous equations at once.

Formulae

y = mx + c or ax + by = d

To represent a linear relationship between two variables, x and y. Graphically, this is a straight line.

y = ax2 + bx + c

To represent a quadratic relationship. Graphically, this is a parabola. A system involving one of these and one linear equation is a linear-quadratic system.

Definitions

Simultaneous Equations
A set of two or more equations that share variables and are all true at the same time. The solution must satisfy every equation in the set.
Elimination Method
An algebraic method for solving simultaneous linear equations where you add or subtract the equations (often after multiplying one or both) to eliminate one of the variables.
Substitution Method
An algebraic method where one equation is rearranged to express one variable in terms of the other, and this expression is then substituted into the second equation.

Worked example

The sum of two numbers is 8. The sum of the squares of the two numbers is 34. What are the two numbers?

  1. 1

    First, translate the problem into two equations.

    Let the numbers be x and y.

  2. 2

    Equation 1 (sum):

    x + y = 8
  3. 3

    Equation 2 (sum of squares):

    x2 + y2 = 34
  4. 4

    This is a linear-quadratic system.

    Use the substitution method.

    Rearrange the linear equation:

    y = 8 - x
  5. 5

    Substitute this expression for y into the quadratic equation:

    x2 + (8 - x)2 = 34.

  6. 6

    Expand the bracket:

    x2 + (64 - 16x + x2) = 34.

  7. 7

    Collect like terms and rearrange into standard quadratic form (ax2 + bx + c = 0):

    2x2 - 16x + 64 = 34 → 2x2 - 16x + 30 = 0.

  8. 8

    Simplify the quadratic by dividing all terms by 2:

    x2 - 8x + 15 = 0.

  9. 9

    Factorise the quadratic:

    (x - 5)(x - 3) = 0.

  10. 10

    This gives two possible values for x:

    x = 5 or x = 3
  11. 11

    Find the corresponding y value for each x.

    If x = 5, y = 8 - 5 = 3
    If x = 3, y = 8 - 3 = 5
  12. 12

    The two solutions are the pair of numbers (5, 3) or (3, 5), which represent the same two numbers.

Answer: The two numbers are 3 and 5.

Common mistakes

  • ×Simple arithmetic errors are the most common mistake. When rearranging equations or expanding brackets, work step-by-step and double-check your calculations.
  • ×Sign errors are frequent, especially when substituting an expression. Squaring a term like (8 - x) must be done carefully: (8 - x)2 = 64 - 16x + x2, not 64 - x2.
  • ×Forgetting to find all solution pairs. If you solve for x and get two values (e.g., x1 and x2), you must substitute both back to find their corresponding y-values (y1 and y2). This system has two (x,y) solution pairs.

No-calculator tips

  • ✓When solving by elimination, multiply equations by the smallest possible integers to create matching coefficients. This keeps the numbers manageable.
  • ✓After solving algebraically, perform a quick mental check. For the worked example, are 3 + 5 = 8? Yes. Is 32 + 52 = 9 + 25 = 34? Yes. This simple check catches most arithmetic slips.
  • ✓When solving a quadratic like `ax2 + bx + c = 0`, if `a` is not 1, always check if the whole equation can be divided by `a` to simplify the numbers before attempting to factorise.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 1 topics