91Ó°ÊÓ

The elimination method is a powerful technique used to solve systems of equations. It involves eliminating one of the variables by adding or subtracting equations from each other. This method is particularly useful when dealing with linear equations because it can simplify the problem significantly.

### Why Use the Elimination Method?- **Efficient**: This method can be faster than substitution, especially when the equations are already aligned properly.- **Clear**: It provides a straightforward approach to remove a variable, making it easier to solve for the other.

### Steps in the Elimination Method1. **Align the Equations**: Make sure the equations are in standard form, i.e., both equations are written as \(ax + by = c\).2. **Choose a Variable to Eliminate**: Decide whether to eliminate \(x\) or \(y\) depending on which can be made to have the same coefficient across both equations with multiplication.3. **Equalize the Coefficients**: Multiply one or both equations so that the coefficients of the chosen variable are equal.4. **Add or Subtract**: Add or subtract the equations to eliminate one variable.5. **Solve for the Remaining Variable**: Once one variable is eliminated, solve for the other.

By using these steps, as demonstrated in this exercise, the system of equations becomes easier to manage and solve.

Linear equations form the foundation of algebra and describe a straight line when graphed on a coordinate plane. A linear equation in two variables \(x\) and \(y\) takes the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

### Characteristics of Linear Equations- **Straight Line Graph**: Represented on a graph as a straight line.- **Constant Rate of Change**: The slope \(m = -\frac{a}{b}\) measures how steep the line is.- **No Exponents**: Linear equations do not have exponent terms (e.g., \(x^2\), \(y^3\)).

### Uses of Linear EquationsLinear equations are used in various real-life scenarios, such as predicting trends, calculating distances, and determining the optimal price point in economics. They are crucial for anyone learning algebra as they underpin the skills needed to solve more complex mathematical problems.

Understanding linear equations is key to mastering systems of equations, as each equation in the system reflects the relationship between the variables involved.

When solving systems of equations, the goal is to find the values of the variables that satisfy all equations simultaneously. In this context, we focus on systems of linear equations.

### Approaches to Solve Systems of Equations - **Graphing**: Plotting each equation on a graph to find the point(s) where they intersect. - **Substitution**: Solving one equation for a variable and then substituting this in the other equation. - **Elimination**: The method used in this exercise, where one variable is eliminated by addition or subtraction.

### Potential Outcomes - **One Solution**: The lines intersect at a single point, indicating one solution (e.g., our ordered pair (5, 10)). - **No Solution**: The lines are parallel and never intersect. - **Infinitely Many Solutions**: The lines are identical, meaning they overlap completely.

Solving systems of equations is an essential skill in algebra, helping students connect mathematical theory with practical problem-solving. By understanding different methods, like elimination, students can approach such problems with confidence and logical precision.