The polynomial degree is the highest variable power in a polynomial.
. In this polynomial, the highest variable power is \(3\).
Important!
It must be noted that there will be a maximum of \(2\) terms in a linear polynomial, \(3\) terms in quadratic polynomials and \(4\) terms in the cubic polynomial of polynomials in one variable.
General form of polynomials of different degrees:
- Linear Polynomial: A polynomial in one variable with degree one is called a linear polynomial. It can be denoted as .
Example:
- Quadratic Polynomial: A polynomial in one variable with degree two is called a quadratic polynomial. It can be denoted as .
Example:
- Cubic Polynomial: A polynomial in one variable with degree three is called a cubic polynomial. It is denoted as .
Example:
A linear equation in which two variables are involved in which each variable is in the first degree.
It can be written in the form \(ax + by + c = 0\) where \(a\), \(b\), \(c\) are real numbers, both \(a\) and \(b\) are not equal to zero, \(x\) and \(y\) are variables and \(c\) is a constant.
It can be written in the form \(ax + by + c = 0\) where \(a\), \(b\), \(c\) are real numbers, both \(a\) and \(b\) are not equal to zero, \(x\) and \(y\) are variables and \(c\) is a constant.
Example:
\(2x + y = 8\), \(x - y - 1 = 0\), \(y = 2x\) are examples of linear equations in two variables.
Substitution method:
Let us learn how to solve the system of linear equations using the substitution method.
The steps to solve the system of linear equations using the substitution method is given by:
Step 1: From the given two equations, consider an equation. Find the value of one of the variables in terms of the other.
Step 2: Substitute the value (obtained in Step 1) in the other equation.
Step 3: Now, simplify the equation and find the value of the other unknown variable.
Step 4: Substituting the obtained value (variable in Step 3) in the equation (obtained in Step 1), we get the value of the first unknown variable.
To understand the concept clearly, let us consider an example.
Example:
Solve the simultaneous linear equations \(2x + y = 8\) and \(x + 2y = 10\) by substitution method.
Solution:
Step 1: Consider an equation and find the value of one of the variables in terms of the other.
Consider the equation \(2x + y = 8\)
Subtract \(2x\) from both the sides, we get:
\(2x + y - 2x = 8 - 2x\)
\(y = 8 - 2x\)
Step 2: Substitute the value of \(y\) in the other equation.
\(x + 2(8 - 2x) = 10\) (Substituting \(y = 8 - 2x\))
Step 3: Solve the above equation and find the value of the variable \(x\).
\(x + 16 - 4x = 10\)
\(-3x + 16 = 10\)
\(-3x = 10 - 16\)
\(-3x = -6\)
\(x = 2\)
Therefore, the value of one of the variables is \(x = 2\).
Step 4: Substitute the value of \(x\) in the equation obtained in step 1.
Thus, substituting \(x = 2\) in the equation \(y = 8 - 2x\), we have:
\(y = 8 - 2(2)\)
\(y = 8 - 4 = 4\)
Therefore, the value of the other variable is \(y = 4\).
Hence, the solution of the given system of equations is \(x = 2\) and \(y = 4\).
Verification:
Substitute the solutions \(x = 2\) and \(y = 4\) in equation (\(1\)), we get:
\(\text{LHS} = 2(2) + 4 = 4 + 4 = 8 = \text{RHS}\)
Similarly, substituting \(x = 2\) and \(y = 4\) in equation (\(2\)), we have:
\(\text{LHS} = 2 + 2(4) = 2 + 8 = 10 = \text{RHS}\)
Since in both the equations \(\text{LHS} = \text{RHS}\), \(x = 2\) and \(y = 4\) is the solution to the given equations.
1. In any of the steps, if we get a false equation like \(0 = 1\), then the system has no solution.
2. If we get an equation like \(0 = 0\), then the system has infinitely many solutions.
Elimination method:
The Elimination method is another method of solving the system of linear equations.
The steps to solve the system of linear equations is given by:
Step 1: First, multiply either one equation or both the equations by a number such that in both the equations, either the coefficients of the first variable or the coefficients of the second variable are equal.
Step 2: Add both the equations or subtract one equation from the other so that the equal coefficients gets cancelled.
Step 3: Solve the obtained equation in Step 2 to find the value of one of the variables.
Step4: Substitute the value of one of the variables(obtained in Step 3) in any one of the two equations to find the value of the remaining variable.
Let us consider an example to understand the concept clearly.
Example:
Solve the simultaneous linear equations \(2x+y = 8\) and \(x+2y = 10\) by elimination method.
Solution:
\(2x+y = 8\) ---- (\(1\))
\(x+2y = 10\) ---- (\(2\))
Step 1: Let us multiply equation (\(1\)) by number \(2\).
Thus, we have:
\(4x+2y =16\)
Step 2: Now, subtract equation (\(2\)) from equation (\(1\)) and cancel the equal coefficients of \(y\).
Step 3: Solve the equation to find the value of the variable \(x\).
\(3x = 6\)
\(x = 2\)
Thus, the value of the variable \(x\) is \(2\).
Step 4: Substitute the value of \(x\) in equation (\(2\)).
Thus, we have:
\(2+2y = 10\)
\(2y = 10-2\)
\(2y = 8\)
\(y = 4\)
Thus, the value of the variable \(y\) is \(4\).
Therefore, the solution of the given system of equations is \(x = 2\) and \(y = 4\).
Important!
1. In any of the steps, if we get a false equation like \(0 = 1\), then the system has no solution.
2. If we get an equation like \(0 = 0\), then the system has infinitely many solutions.