JENECONKMaths Universe

Matrices and Simultaneous Equations

Ages 14–15

Difficulty adapts to you

Matrices and Simultaneous Equations

Two linear equations can be written as AX = B. The coefficient matrix A contains the coefficients of x and y, X is the column matrix [[x], [y]], and B contains the constants. If A is non-singular, multiply both sides by A⁻¹ to get X = A⁻¹B. Always substitute the values back into both original equations to check the solution.

Your goals

  • ✓You can represent and solve two linear equations using matrix form and an inverse.
  • ✓You can use coefficient matrix, unknown vector, unique solution correctly.
  • ✓You can keep the variable order the same in every row and substitute back into both equations.

Words to know

coefficient matrix:
The matrix containing the coefficients of the unknowns.
unknown vector:
A column matrix containing the variables to find.
unique solution:
One pair of values satisfying every equation.

Examples

  • 12x + y = 7 and x − y = 1 become 211−1 xy = 71.
  • 2For x + y = 9 and x − y = 3, adding the equations gives 2x = 12, so x = 6 and y = 3; matrix and elimination methods agree.
  • 3If the coefficient determinant is 0, the equations do not have a unique solution: they may be parallel (no solution) or the same line (infinitely many solutions).
  • 4The order of coefficients must match the variable order. If X is xy, the first column is the x-coefficient and the second is the y-coefficient.
  • 5Check (6, 3): 2(6) + 3 = 15, so it would not solve a system whose first constant is 7. Substitution catches copied signs and order errors.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Build AX = B

Best for: Translating equations into matrix form
  1. 1.Put coefficients in A, in variable order.
  2. 2.Put unknowns in the column X.
  3. 3.Put constants in B.

Example: 2x+y=7 and x−y=1 become [[2,1],[1,-1]][[x],[y]]=[[7],[1]].

Method 2

Inverse then verify

Best for: Solving and checking a system
  1. 1.Check det(A) is not zero.
  2. 2.Calculate X = A⁻¹B.
  3. 3.Substitute x and y into both original equations.

Example: A unique solution must satisfy both equations, not just one of them.

Check these

  • • Do not place constants in the coefficient matrix or swap the x and y columns.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own matrices and simultaneous equations example, solve it, and explain how you checked it.

Byte

Byte

Helpful Hints

“Use your fingers or objects to count — it helps!”