Simultaneous Linear Equations • Topic 2 of 3

Method of Cross-Multiplication

What is the Method of Cross-Multiplication?

The Method of Cross-Multiplication is a structured, formula-driven method used to solve simultaneous linear equations. Instead of deciding whether to add, subtract, or substitute, this method provides a fixed matrix-like diagrammatic formula where you directly plug in the numerical coefficients to find the values of x and y immediately.

To use this method properly, both equations must first be written in their standard, uniform layout. The standard format requires all variables and constant numbers to sit on the left side of the equals sign, leaving exactly zero on the right side:

Equation 1: a1(x) + b1(y) + c1 = 0

Equation 2: a2(x) + b2(y) + c2 = 0

Here, the letters a, b, and c represent the fixed real number numbers (coefficients and constants) from the equations. The indices 1 and 2 simply track which equation the number came from. The cross-multiplication formula states that the ratios of the variables to the cross-products of their non-corresponding coefficients are completely equal.

Substitution MethodSolve: x + 2y = 7 ... (1) 3x − y = 5 ... (2)Step 1(1)From (1): x = 7 − 2yStep 2(2)Substitute in (2): 3(7−2y) − y = 5Step 3(3)21 − 6y − y = 5 → 7y = 16 → y = 16/7...Step 4(4)x = 7 − 2y = 7 − 2(16/7) = 49/7 − 32/7 = 17/7Check(5)x+2y=17/7+32/7=49/7=7 ✓ 3x−y=51/7−16/7=35/7=5 ✓Best used when: one equation already gives a variable explicitlye.g. y = 3x + 2 or x = 5 − y (coefficient of 1)
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Worked Example

Solve a standard problem on Method of Cross-Multiplication.

Solution

Apply the formula/method shown in the concept section above.

Key Points

  • Understand the definition and properties of Method of Cross-Multiplication.
  • Study the worked examples and practice similar problems.
  • Always verify your answer using the original conditions.
Tap an option to check your answer0 / 4
Q1.Equations for cross-multiplication must be written as:
Explanation: Standard form.
Q2.The common denominator for $x$ and $y$ is:
Explanation: $a_1b_2-a_2b_1$.
Q3.If $a_1b_2-a_2b_1=0$, the pair may have no:
Explanation: No unique solution.
Q4.Cross-multiplication gives $x$ and $y$ in terms of the:
Explanation: Coefficients.