Elimination is a method used in algebra to solve a system of linear equations. It involves manipulating the equations in such a way that when they are added or subtracted, one variable is eliminated, allowing for the solution of the remaining variable(s). This method is particularly useful when dealing with systems of equations with two or more variables.
To successfully apply elimination, one must have a solid understanding of the following concepts:
Linear equations: These are equations in which the highest power of the variable is 1. They can be written in the form of "ax + by = c," where "a," "b," and "c" are constants.
Systems of equations: These are a set of two or more equations that share the same variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.
The step-by-step process of elimination involves the following:
Ensure that the equations are written in standard form, with the variables on the left side and the constants on the right side.
Choose one variable to eliminate by multiplying one or both equations by appropriate constants so that the coefficients of that variable in both equations become additive inverses (i.e., they add up to zero when added together).
Add or subtract the equations to eliminate the chosen variable. This will result in a new equation with only one variable.
Solve the new equation for the remaining variable.
Substitute the value of the solved variable back into one of the original equations to find the value of the eliminated variable.
Check the solution by substituting the values of the variables into all the original equations. If the solution satisfies all the equations, it is the correct solution.
The elimination method does not have a specific formula or equation. It is a process that involves manipulating the given equations to eliminate one variable and solve for the remaining variable(s).
As mentioned earlier, there is no specific formula or equation for elimination. Instead, the method involves manipulating the given equations to eliminate one variable. This is done by multiplying one or both equations by appropriate constants so that the coefficients of the chosen variable become additive inverses. The equations are then added or subtracted to eliminate the chosen variable.
There is no specific symbol for elimination in mathematics. It is simply referred to as the "elimination method" or "elimination."
There are two common methods for elimination:
Addition/Subtraction Method: In this method, the equations are added or subtracted to eliminate one variable. The goal is to manipulate the equations so that the coefficients of the chosen variable become additive inverses. Once the variable is eliminated, the remaining equation can be solved for the remaining variable.
Multiplication Method: In this method, one or both equations are multiplied by appropriate constants to make the coefficients of the chosen variable additive inverses. This allows for the elimination of the variable when the equations are added or subtracted.
Both methods are equally valid and can be used interchangeably depending on the given equations and personal preference.
Example 1: Solve the following system of equations using the elimination method:
Equation 1: 2x + 3y = 7 Equation 2: 4x - 2y = 10
Step 1: Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of "x" additive inverses: 2(2x + 3y) = 2(7) -> 4x + 6y = 14 3(4x - 2y) = 3(10) -> 12x - 6y = 30
Step 2: Add the two equations to eliminate "y": (4x + 6y) + (12x - 6y) = 14 + 30 16x = 44
Step 3: Solve for "x": 16x = 44 -> x = 44/16 -> x = 11/4
Step 4: Substitute the value of "x" back into Equation 1 to find "y": 2(11/4) + 3y = 7 11/2 + 3y = 7 3y = 7 - 11/2 3y = 14/2 - 11/2 3y = 3/2 y = 1/2
Therefore, the solution to the system of equations is x = 11/4 and y = 1/2.
Example 2: Solve the following system of equations using the elimination method:
Equation 1: 3x - 2y = 8 Equation 2: 2x + 5y = 1
Step 1: Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of "x" additive inverses: 2(3x - 2y) = 2(8) -> 6x - 4y = 16 3(2x + 5y) = 3(1) -> 6x + 15y = 3
Step 2: Subtract the two equations to eliminate "x": (6x - 4y) - (6x + 15y) = 16 - 3 -19y = 13
Step 3: Solve for "y": -19y = 13 -> y = 13/-19 -> y = -13/19
Step 4: Substitute the value of "y" back into Equation 1 to find "x": 3x - 2(-13/19) = 8 3x + 26/19 = 8 3x = 8 - 26/19 3x = 152/19 - 26/19 3x = 126/19 x = 42/19
Therefore, the solution to the system of equations is x = 42/19 and y = -13/19.
Solve the following system of equations using the elimination method: Equation 1: 5x + 2y = 11 Equation 2: 3x - 4y = 7
Solve the following system of equations using the elimination method: Equation 1: 2x - 3y = 5 Equation 2: 4x + 5y = 1
Solve the following system of equations using the elimination method: Equation 1: 6x + 7y = 8 Equation 2: 3x - 2y = 5
Question: What is the difference between elimination and substitution methods?
Answer: The elimination method involves manipulating the equations to eliminate one variable by adding or subtracting the equations. On the other hand, the substitution method involves solving one equation for one variable and substituting that expression into the other equation. Both methods are used to solve systems of equations, but the elimination method is often preferred when dealing with equations that have coefficients that are easily manipulated to become additive inverses.