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CRAMER'S RULE FOR 2x2 SYSTEMS
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.
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CRAMER'S RULE FOR 2x2 SYSTEMS
What is it?
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.
In simple words
- For ax+by=e, cx+dy=f, the coefficient matrix A=[[a,b],[c,d]].
- THE MATRICES: 2x+3y=7, x+4y=6: A=[[2,3],[1,4]].
- DIVIDE: |A|=5, |Ax|=10, |Ay|=5 so x=2, y=1.
- Worked-method reminder: check: 1(2)+4(1)=6
Important terms
CRAMER'S RULE FOR 2x2 SYSTEMS: 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.
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.
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).
What you should learn
- 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.
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
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.
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).
Recommended for Class 10
Full school-level explanation
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.
Det of A, then two more dets:
WHAT — CRAMER'S RULE FOR 2x2 SYSTEMS
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. The symbols and conditions are part of the mathematics, so include them when stating the rule.
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.
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.
Remember
- constants into one column
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).
Remember
- divide by |A|
Examples
- x=105=2
- y=55=1
- Check: 4+3=7, 2+4=6
Remember
- x = |Ax|/|A|
- replace one column
- x=2, y=1
- one column at a time
- constants into one column
- divide by |A|
- x = |Ax|/|A|
- replace one column
How the ideas connect
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
Deeper understanding
Go beyond memorising CRAMER'S RULE FOR 2x2 SYSTEMS. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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
Common mistakes
Exam focus
Core Idea
THE MATRICES
DIVIDE
Worked Method
Exam focus
Quick revision
- x = |Ax|/|A|
- replace one column
- x=2, y=1
- one column at a time
- CRAMER'S RULE FOR 2x2 SYSTEMS: For ax+by=e, cx+dy=f, the coefficient matrix A=[[a,b],[c,d]]. x = |Ax|/|A| and y = |Ay|/|A|, where
- 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 coe
Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.
For long answers on CRAMER'S RULE FOR 2x2 SYSTEMS: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- x = |Ax|/|A|
- replace one column
- x=2, y=1
- one column at a time
Important formulas
- x = |Ax|/|A|
- replace one column
- x=2, y=1
Common exam traps
- Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.
- Do not treat THE MATRICES and DIVIDE as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.
- Mistake
Do not treat THE MATRICES and DIVIDE as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
x = |Ax|/|A|
replace one column
x=2, y=1
one column at a time
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 replace
x = |Ax|/|A|
- 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 replace
- Formula
- x = |Ax|/|A|
- Example
- A from x and y coeffs
- Critical warning
- Do not replace both columns at once. Ax keeps the y-coefficients; Ay keeps the x-coefficients.
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Exam Practice
Define
Q1. Define CRAMER'S RULE FOR 2x2 SYSTEMS.
Explain
Q2. Explain: THE MATRICES.
Example
Q3. Give one clear example of CRAMER'S RULE FOR 2x2 SYSTEMS and say why it fits.
Mistake
Q4. State one common mistake students make about CRAMER'S RULE FOR 2x2 SYSTEMS.
Exam note
Q5. Write a short exam note on CRAMER'S RULE FOR 2x2 SYSTEMS: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.