What is an expression?
Expressions are mathematical statements that contain at least two terms containing numbers or variables, or both, and are connected by an operator. Addition, subtraction, multiplication, and division are some examples of mathematical operators.
For example, 3x + 5 is an expression in which 3x and 5 are terms separated by an addition operator.
What are equations?
Equations are mathematical statements that have algebraic expressions on both sides of an equal sign (=). It depicts the equality relationship between the expressions written on the left-hand side and the expressions written on the right-hand side. L.H.S = R.H.S (left hand side = right hand side) appears in every math equation.
Check whether these are expressions or equations:
Equations | Is it an equation? | |
|---|---|---|
1. | y = 3 x + 5 | equation |
2. | 7 + 2 = 10 - 1 | equation |
3. | \(y+x^2+4\) | expression |
Linear Equation
Linear equations are mathematical equations with 1 as the highest degree of the variables. In other words, the highest exponent of the terms in such equations is 1. These can be further divided into one-variable linear equations, two-variable linear equations, three-variable linear equations, and so on.
A linear equation with variables x and y has the standard form ax + by – c = 0, where a and b are the coefficients of x and y, and c is the constant.
Solving Linear Equations by Elimination Method
One of the methods of solving a system of linear equations is the elimination method. To get the equation in one variable using this method, we can either add or subtract the equations. We can add the equations to eliminate a variable if the coefficients of one of the variables are the same, and the sign of the coefficients is the opposite.
We can also subtract the equations to get the equation in one variable, if the coefficients of one of the variables are the same, and the sign of the coefficients is the same.
If we don’t have the equations in a form to directly add or subtract the equations to eliminate the variable, we can start by multiplying one or both equations by a constant value on both sides of the equation.
We do this to get an equivalent linear system of equations, after which, we can eliminate the variable by simply adding or subtracting the equations.
Steps in the Elimination Method
Step 1: To make the coefficients of any one of the variables (either x or y) numerically equal, multiply both the given equations by some suitable non-zero constants. TIP: Correlate this with the Least Common Multiple (LCM).
Step 2: After that, add or subtract one equation from the other so that one variable is eliminated.
Step 3: To find the value of one variable (x or y), solve the equation in one variable (x or y).
Step 4: To get the value of another variable, substitute this value into any of the given equations.
Solved System of Linear Equation Examples by Elimination Method
Example 1:
Solve the system by elimination.
x + 5y = 2 Equation 1.
x – 5y = 16 Equation 2.
Solution:
Step 1: The coefficients of the y-terms are opposite in terms of the signs, as you can see. As a result, you can add the equations to get a single variable equation for x.
x + 4y = 2 Equation 1.
2x – 4y = 16 Equation 2.
3x = 18 Add the equations.
Step 2: Solve for x.
3x = 18 Equation from step 1.
x = 6 Divide each side by 3.
Step 3: Substitute 6 for x in one of the original equations and solve for y.
x + 4y = 2 Equation 1
6 + 4y = 2 Substitute 6 for x.
4y = -4 Subtract 4 from each side.
y = -1 Divide each side by 4.
Hence, the required solution is (6,-1).
Example 2:
Solve the system by elimination.
-8x + 5y = 25 Equation 1
-2x – 4y = 14 Equation 2
Solution:
Step 1: It’s important to note that no two similar terms have the same or opposite coefficients. Multiply equation 2 by 4, so that the x terms have a coefficient of -8.
-8x + 5y = 23 -8x + 5y = 23 Equation 1
-2x-3y=10 Multiply by 4. -8x – 12y = 40 Revised equation 2
Step 2: Subtract the equations to obtain an equation in one variable, y.
-8x + 5y = 23 Equation 1.
-8x – 12y = 40 Revised equation 2.
17y = -17 Subtract the equations.
Step 3: Solve for y.
17y = -17 Equation from step 2
y = -1 Divide each side by 17.
Step 4: Substitute -1 for y in one of the original equations and solve for x.
-2x – 3y = 10 Equation 2.
-2x – 3 – 1 = 10 Substitute -1 for y.
-2x + 3 = 10 Multiply.
-2x = 7 Subtract 3 from each side.
x = –72 = -3.5 Divide each side by -2.
Hence, the required solution is (-3.5,-1).
The elimination method is a method for removing one of the variables from a system of linear equations by using addition or subtraction in conjunction with multiplication or division of the coefficients of the variables.
The following are some of the benefits of using the elimination method:
1. There are fewer steps in the elimination method than in other methods.
2. When compared to other methods, it reduces the risk of making a mistake while solving a problem.
The following are the various methods for solving a system of linear equations:
1. Substitution Method
2. Graphical Method
3. Elimination Method