Solving systems of equations by elimination is a powerful technique in algebra. This method allows you to find solutions to complex problems by strategically combining equations to cancel out variables.
Elimination is particularly useful when dealing with real-world scenarios involving multiple unknowns. By mastering this technique, you'll be able to tackle a wide range of practical problems in business, science, and everyday life.
Solving Systems of Equations by Elimination
Elimination method for equation systems
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Involves adding or to eliminate one
Creates an equation with only one variable that can be easily solved
Steps for solving systems using elimination:
Multiply one or both equations by a constant to make the coefficients of one variable opposites
Choose the variable easiest to eliminate (usually with the smallest coefficients)
Add the equations together to eliminate the chosen variable
Like terms will cancel out, leaving an equation with only one variable
Solve the resulting equation for the remaining variable
Substitute the value of the solved variable into one of the original equations to find the value of the other variable
Check the solution by substituting the values into both original equations
Example:
Given the : 3x+2y=11 and 2x−2y=2
Multiply the second equation by -1 to get −2x+2y=−2
Add the equations: 3x+2y=11 and −2x+2y=−2 to get x+4y=9
Solve for y to get y=2
Substitute y=2 into 3x+2y=11 to solve for x and get x=1
Solution: x=1, y=2
Real-world applications of elimination
Identify unknown quantities and assign variables (x and y)
Create a system of equations based on given information
Each equation represents a relationship between unknown quantities
Use to solve the system of equations
Follow steps outlined in previous objective
Interpret solution in context of real-world problem
Ensure solution makes sense and answers original question
Example:
A small business sells two types of gift baskets: regular and deluxe
The regular basket costs 30andthedeluxebasketcosts50
The business sold a total of 20 baskets and made $700 in revenue
Let x = number of regular baskets and y = number of deluxe baskets
System of equations: x+y=20 and 30x+50y=700
Solve using elimination to get x=15 and y=5
Interpretation: The business sold 15 regular baskets and 5 deluxe baskets
Efficiency of elimination vs other methods
Elimination method often most efficient when:
Coefficients of one variable are opposites or easily made opposites by multiplication
Coefficients are small integers
may be more efficient when:
One equation has a variable with of 1 or -1
Equations are already solved for one variable
may be more efficient when:
Equations are in (y=mx+b)
Visual representation of solution is desired
In some cases, combination of methods may be most efficient
Use substitution to solve for one variable, then use elimination to solve for the other
Example:
Given the system of equations: y=2x+1 and 4x+2y=14
Substitution is efficient since first equation is solved for y
Substitute y=2x+1 into 4x+2y=14 to get 4x+2(2x+1)=14
Simplify and solve for x to get x=2
Substitute x=2 into y=2x+1 to get y=5
Solution: x=2, y=5
Algebraic Techniques in Elimination
: Another term for systems of equations that are solved together
: The process of modifying equations to create equivalent forms
Used to prepare equations for elimination by making coefficients opposites
: The result of adding or subtracting terms with opposite signs, eliminating variables
: The method of by constants and then adding them to create a new equation