Let us recall the concepts in Algebra:
1. A system of linear equations in three variables will be according to one of the following cases. (i) Unique solution (ii) Infinitely many solutions (iii) No solution.
Steps to solve the system of linear equations in three variables:
Step 1: Consider any \(2\) equations from \(3\) equations. Multiply the equations with suitable values such that any one of the variables gets cancelled, leaving the variables either \(x\) and \(y\) or \(x\) and \(z\) or \(y\) and \(z\).
Step 2: Again, consider any \(2\) equations from \(3\) equations and eliminate the same variable which was eliminated in the previous pair.
Step 3: Now, we have \(2\) equations with two variables.
Step 4: Solve these \(2\) equations using any method like substitution or elimination or cross multiplication method.
Step 5: Substitute the value of these \(2\) variables in any of the given equations and determine the value of the third variable.
Step 2: Again, consider any \(2\) equations from \(3\) equations and eliminate the same variable which was eliminated in the previous pair.
Step 3: Now, we have \(2\) equations with two variables.
Step 4: Solve these \(2\) equations using any method like substitution or elimination or cross multiplication method.
Step 5: Substitute the value of these \(2\) variables in any of the given equations and determine the value of the third variable.
Important!
1. In any of the steps, the system is inconsistent and has no solution if we get a false equation like \(0 = 1\),
2. If we get an equation like \(0 = 0\), the system is consistent and has infinitely many solutions.
2. LCM of polynomials: The Least Common Multiple of two or more algebraic expressions is the expression of the lowest degree (or power), divisible by each of them without remainder.
The LCM of a number or an algebraic expression by factorization method can be determined using the following steps:
(i) Each expression must be resolved into its simple factors.
(ii) The highest power of the common factors will be the LCM.
(iii) If the expressions have numerical coefficients, find their LCM.
(iv) The product of the LCM of factors and coefficient is the required LCM.
(i) Each expression must be resolved into its simple factors.
(ii) The highest power of the common factors will be the LCM.
(iii) If the expressions have numerical coefficients, find their LCM.
(iv) The product of the LCM of factors and coefficient is the required LCM.
3. GCD of polynomials: GCD of polynomials can be determined using long division method.
Let \(f(x)\) and \(g(x)\) be two polynomial such that \(deg(f(x)) \geq deg(g(x))\). Then, the divisor is \(g(x)\).
The steps to find the Greatest Common Divisor(GCD) or the Highest Common Factor(HCF) of two polynomials \(f(x)\) and \(g(x)\) is given by:
Step 1: Divide \(f(x)\) by \(g(x)\) to obtain \(f(x) = g(x)q(x) + r(x)\) where \(q(x)\) is the quotient and \(r(x)\) is the remainder. Then, \(deg[r(x)] < deg[g(x)]\). If the remainder \(r(x)\) is zero, then \(g(x)\) is the GCD of \(f(x)\) and \(g(x)\).
Step 2: If the remainder \(r(x)\) is a non - zero, divide \(g(x)\) by \(r(x)\) such that \(g(x) = r(x)q(x) + r_1(x)\) where \(r_1(x)\) is the new remainder. Then, \(deg[r_1(x)] < deg[r(x)]\). If the remainder \(r_1(x)\) is zero, then \(r(x)\) is the GCD.
Step 3: If \(r_1(x)\) is non - zero, repeat step \(2\) until we get the remainder zero.
4. Relationship between LCM and GCD of polynomials is the product of any two polynomials are equal to the product of their LCM and GCD.
That is, \(f(x) \times g(x) = LCM [f(x), g(x)] \times GCD [f(x), g(x)]\).
5. An expression is called a rational expression if it can be written in the form \(\frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \neq 0\).
Reduction of Rational Expression
A rational expression \(\frac{p(x)}{q(x)}\) is said to be in its lowest form if the greatest common divisor of \(p(x)\) and \(q(x)\) is \(1\).
That is \(GCD \left(p(x), q(x) \right)\) = \(1\).
Working rule to reduce a rational expression to its lowest form:
Step 1: Simplify or factorise the numerator \(p(x)\) and the denominator \(q(x)\).
Step 2: Cancel out the common factors in the numerator and the denominator.
Step 3: The final expression obtained after the above two steps is the rational expression in its lowest form.
6. A value that makes a rational expression (in its lowest form) undefined is called an Excluded value.
Working rule to find the excluded value of a rational number:
Step 1: Simplify or factorise the numerator \(p(x)\) and the denominator \(p(x)\).
Step 2: Cancel out the common factors in the numerator and the denominator.
Step 3: Equate the lowest form of the denominator \(q(x)\) to zero.
Step 4: Thus, the obtained value for which the denominator becomes zero is the excluded value of that rational number.