๐ Understanding Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This allows you to solve for the remaining variable. Itโs super handy when one equation is already solved for a variable or can easily be rearranged.
๐ Understanding Elimination Method
The elimination method, also known as the addition method, focuses on eliminating one of the variables by adding or subtracting the equations. To do this, you might need to multiply one or both equations by a constant so that the coefficients of one variable are opposites. This method is particularly useful when the coefficients of one variable are easily made opposites.
๐ Substitution vs. Elimination: A Detailed Comparison
| Feature |
Substitution Method |
Elimination Method |
| Definition |
Solving one equation for one variable and substituting it into the other. |
Eliminating one variable by adding or subtracting the equations. |
| Best Use Case |
When one equation is easily solved for a variable. |
When the coefficients of one variable are easily made opposites. |
| Steps |
- Solve one equation for one variable.
- Substitute the expression into the other equation.
- Solve for the remaining variable.
- Substitute back to find the other variable.
|
- Multiply one or both equations to make coefficients opposites.
- Add or subtract the equations to eliminate a variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
|
| Example |
Solve the system:
$y = 2x + 1$
$3x + y = 10$
Substitute $y$ in the second equation:
$3x + (2x + 1) = 10$
|
Solve the system:
$x + y = 5$
$x - y = 1$
Add the equations to eliminate $y$:
$2x = 6$
|
๐ก Key Takeaways
- ๐ Substitution: Works best when one variable is already isolated or can be easily isolated.
- โ Elimination: Works best when the coefficients of one variable are easily made opposites or identical.
- ๐ง Choosing the Right Method: Consider the structure of the equations. If one equation is already solved for a variable, substitution is likely easier. If the coefficients are easily manipulated to be opposites, elimination is a good choice.
- ๐ Flexibility: Both methods will lead to the same solution. Sometimes, one method is just more efficient than the other.
- ๐ Practice: The more you practice, the better you'll become at recognizing which method is best suited for a particular system of equations.