MathematicsChapter 3: Matrices and Determinants

CRAMER'S RULE FOR 2x2 SYSTEMS

For ax+by=e, cx+dy=f, the coefficient matrix A=[[a,b],[c,d]].

Learning objectives

Explain cramer's rule for 2x2 systems using the correct definitions, conditions, and notation.
Apply x = |Ax|/|A|, replace one column correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

For ax+by=e, cx+dy=f, the coefficient matrix A=[[a,b],[c,d]]. x = |Ax|/|A| and y = |Ay|/|A|, where Ax replaces the x-column by the constants, and Ay replaces the y-column. Need |A| != 0. This is a school method, not a computer program.

Mathematical form

x = 105 = 2

y = 55 = 1

by=e

dy=f

Where / symbols

CRAMER'S RULE FOR 2x2 SYSTEMS: For ax+by=e, cx+dy=f, the coefficient matrix A=[[a,b],[c,d]]. x = |A_x|/|A| and y = |A_y|/|A|, where A_x replaces the x-column by the constants, and A_y replaces the y-column. Need |A| != 0. This is a school method, not a computer program.
THE MATRICES: 2x+3y=7, x+4y=6: A=[[2,3],[1,4]]. A_x=[[7,3],[6,4]]. A_y=[[2,7],[1,6]]. Keep the other column as coefficients.
DIVIDE: |A|=5, |Ax|=10, |Ay|=5 so x=2, y=1. Check in both original equations. If |A|=0 the method stops (no unique solution).

Relationship

x = |Ax|/|A|

Overview

Det of A, then two more dets:

The Matrices

2x+3y=7, x+4y=6: A=[[2,3],[1,4]]. Ax=[[7,3],[6,4]]. Ay=[[2,7],[1,6]]. Keep the other column as coefficients.

Examples A from x and y coeffs · A_x puts 7 and 6 in col 1 · A_y puts 7 and 6 in col 2

Divide

|A|=5, |Ax|=10, |Ay|=5 so x=2, y=1. Check in both original equations. If |A|=0 the method stops (no unique solution).

Examples x=10/5=2 · y=5/5=1 · Check: 4+3=7, 2+4=6

WHAT — CRAMER'S RULE FOR 2x2 SYSTEMS

WHY — Why this topic matters

Cramer'S Rule For 2X2 Systems matters because matrices organise numbers and give short methods for transformations and simultaneous equations. Correct order, determinant, and multiplication rules prevent most exam errors.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through THE MATRICES, then DIVIDE. Worked: 2x + 3y = 7, x + 4y = 6. |A| = 24 - 31 = 5 |Ax| = 74 - 36 = 28 - 18 = 10 x = 105 = 2 |Ay| = 26 - 71 = 12 - 7 = 5 y = 55 = 1 check: 2(2)+3(1)=7 check: 1(2)+4(1)=6 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

2x + 3y = 7, x + 4y = 6.
|A| = 2*4 - 3*1 = 5
|Ax| = 7*4 - 3*6 = 28 - 18 = 10
x = 105 = 2
|Ay| = 2*6 - 7*1 = 12 - 7 = 5
y = 55 = 1
check: 2(2)+3(1)=7
check: 1(2)+4(1)=6

In simple words

Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.

Key points

x = |Ax|/|A|
replace one column
constants into one column
divide by |A|

Exam preparation

Exam Focus

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Review the key facts, formulas and checks before you move on.