Algebraic methods provide a systematic approach for solving a pair of linear equations in two variables. These methods involve manipulating and combining the eq...
Algebraic methods provide a systematic approach for solving a pair of linear equations in two variables. These methods involve manipulating and combining the equations to obtain a single, linear equation that represents both equations simultaneously.
The process typically involves isolating one variable in one equation and then substituting it into the other equation. This process helps to eliminate the variable with unknown values, simplifying the equations and allowing for solution.
To illustrate the application of these methods, consider the following pair of linear equations:
Eq. 1: 2x + y = 7
Eq. 2: 3x - y = 5
To solve this pair, we can use the method of substitution. First, we isolate the y variable in Eq. 1:
2x = 7 - y
Then, we substitute this expression for y into Eq. 2:
3x - (7 - y) = 5
Simplifying this equation, we get:
4x - 7 = 5
Adding 7 to both sides, we obtain:
4x = 12
Dividing both sides by 4, we find x = 3
Now that we know x = 3, we can substitute this value back into either of the original equations to solve for y. Using Eq. 1, we get:
2(3) + y = 7
Which simplifies to y = 5
Therefore, the solution to the system of equations is x = 3 and y = 5.
This solution demonstrates how algebraic methods allow us to find the values of x and y that simultaneously satisfy both linear equations. By systematically manipulating and combining these equations, we can eliminate the variable with unknown values, resulting in a single, linear equation that represents both equations