# Higher Algebra Hall & Knight Indeterminate Equations of the First Degree (Chapter 26) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 26 ‘Indeterminate Equations of the First Degree’ have been prepared to assist you in solving these advanced level exercise questions effortlessly. These solutions are a perfect guide for IIT JEE and NEET preparation. In this chapter of Higher Algebra by Hall and Knight, you will learn to solve the indeterminate equations of the first degree using the properties of continued fractions learnt in the previous chapter. The topics covered in this chapter are solutions of ax-by=c, finding the general solution using one solution of the equation, solution of ax+by=c, number of solutions in az+by=c. Besides all the above topics, you will also get to know the steps to find the solutions of ax+by+cz=d, a’x+b’y+c’z=d’.

Higher Algebra By H.S. Hall and S.R. Knight Intermediate Equations of First Degree has 24 questions covered in 1 exercise. The questions will help you attain complete clarity in the concepts of indeterminate equations and the different methods to find their solutions. You will get acquainted with the most commonly occurring maths problem in your competitive exams and in your higher studies. The questions can be solved only with good conceptual clarity in the topics. We advise you to take guidance from Instasolv’s solutions for Hall and Knight Higher Algebra book.

The expert team of Math teachers at Instasolv has created this set of solutions with great care using the simple and understandable language for explaining the concepts of Indeterminate Equations of the First Degree. You will find assistance at every step with precise reasoning. The solutions of this classic algebra book have been written in a format strictly compliant to the latest JEE guidelines.

## Important Topics for Hall and Knight Higher Algebra Solutions Chapter 26: Indeterminate Equations of the First Degree

**Introduction**

All the equations in two variables of the first degree can be written in the form axby=c, assuming a, b, and c as positive integers. Such an equation has infinite solutions unless we expect x and y to be positive integers. In that case, solutions are limited.

The only axby=c of the above equations can be considered for integral solutions since if a and b have a factor that isn’t a factor of c, then the above equations would not have any integral solutions.

**General Solution in Positive Integers for ****ax-by=c**

These solutions do not hold true for a or b equal to 1.

**General Solution when one Positive Integral Solution is given for**

*ax – by = c*

Let the solutions are h and k and

Then x = h + bt, and y = k + at, are the general solutions.

**General Solution in Positive Integers for ****ax+by=c**

**General Solution when one Positive Integral Solution is given for ****ax+by=c**

Let the solutions are h and k and

Then x = h + bt, and y = k – *at*, are the general solutions.

**Number of positive integral solutions for ****ax+by=c**

**Solution of ****ax + by + cz = d**

We can rearrange it in the form, ax + by = d – czax + by = c’

And thus, solve using the steps discussed above.

**General Solution of Simultaneous Equations **a x + by + cz = d **and**

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 26: Indeterminate Equations of the First Degree

- In the exercise questions of Higher Algebra by Hall & Knight Intermediate Equations of First Degree, you will get to solve both formulae based as well as concept-based questions
- You will be required to find the least positive integral solution of given equations.
- You will also get questions in which you will need to evaluate the continued fractions, given their denominators.
- Lastly, there are word problems based on the ringing of bells or division of rods in which you will have to find the equation before finding the solutions of the problems.

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