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Logarithm Equations and Cramer's Rule: Solving Simultaneous Equations
Published on June 7, 2026
Algebra
Learn how to solve logarithm equations and systems of linear equations using Cramer's rule and matrix inverse methods. This guide provides step-by-step solutions.
Question
Solve the question attached
Answer
Based on the image provided, here are the step-by-step solutions for the visible math problems.
QUESTION ONE
a) Solve for
i)
This is a statement to verify or solve. Let’s check if it’s true:
By definition of a logarithm,
Here,
The equation is an identity, so it is true for the given base and value.
This is a statement to verify or solve. Let’s check if it’s true:
By definition of a logarithm,
Here,
The equation is an identity, so it is true for the given base and value.
ii)
Convert the logarithmic equation to exponential form:
Convert the logarithmic equation to exponential form:
Express both sides as powers of
Since the bases are equal, the exponents must be equal:
iii)
First, use the product rule
First, use the product rule
Move the log term to the left:
Use the product rule again:
Convert to exponential form:
Use the quadratic formula
Since the argument of a logarithm must be positive (
b) Use Cramer’s rule to solve the following simultaneous equations:
-
-
(Note: The image shows , let’s re-read carefully. It looks like ) -
Let’s re-examine the equations from the image:
Step 1: Find the determinant of the coefficient matrix (
Expand along the third column:
Step 2: Find
Step 3: Find
Step 4: Find
Solution:
QUESTION TWO
a) Ann, Beth and Martin… share profit in the ratio 7:2:4. Total profit is K7.8m. How much profit does [Ann] get?
Total parts
Ann’s share
Ann’s share
Ann’s share
Ann’s share
b) Factorise:
This is a difference of two squares:
Answer:
Answer:
c) Solve the system using inverse method:
Matrix form
Find
Solve for
Solution:
d) Evaluate
Using log properties:
Calculation:
Answer: 1
Calculation:
Answer: 1
QUESTION THREE
a) Half the sum of two numbers is 20 and three times their difference is 18. Find the numbers.
Let the numbers be
Add the equations:
Substitute
The numbers are 23 and 17.
The numbers are 23 and 17.