In Algebra 1, when solving systems of equations, you have several methods at your disposal: substitution, elimination, and equal values. Each method can be used effectively in different situations. Here’s a simple but detailed explanation of when to use each method.
1. Substitution Method
When to Use:
- Use substitution when one of the equations is easily solvable for one variable. This is especially useful if you have a linear equation that isolates a variable (like \( y = mx + b \)).
How It Works:
- Solve one equation for one variable.
- Substitute this expression into the other equation.
- Solve for the remaining variable.
- Back-substitute to find the other variable.
Example: Given the system: \[ y = 2x + 3 \] \[ 3x + 2y = 12 \] You can see that \( y \) is already isolated in the first equation. You can substitute \( y \) in the second equation.
2. Elimination Method
When to Use:
- Use elimination when the coefficients of one of the variables are the same or can be easily manipulated to become the same. This method is especially useful for equations that are already lined up vertically, or when you can quickly multiply one or both equations to align coefficients.
How It Works:
- Align the equations vertically.
- Multiply one or both equations if necessary so that the coefficients of one of the variables are opposites.
- Add or subtract the equations to eliminate one variable.
- Solve for the remaining variable and back-substitute to find the other.
Example: Given the system: \[ 2x + 3y = 6 \] \[ 4x - 3y = 8 \] Here, the coefficients of \( y \) can be eliminated immediately because \( 3y \) and \( -3y \) are opposites.
3. Equal Values Method
When to Use:
- Use the equal values method when both equations are already solving for the same variable, making it straightforward to set them equal to each other.
How It Works:
- Solve both equations for the same variable.
- Set the two expressions equal to each other.
- Solve for the variable, then substitute back to find the other variable.
Example: Given the system: \[ y = 2x + 3 \] \[ y = -x + 5 \] Since both equations are equal to \( y \), you can set \( 2x + 3 \) equal to \( -x + 5 \) to solve for \( x \).
Summary:
- Substitution is best when one variable is easy to isolate.
- Elimination is useful when you can line up the equations and easily eliminate a variable.
- Equal Values is effective when both equations are already expressed in terms of the same variable.
Practice using these techniques with different types of equation systems to get a feel for which method works best in different situations!