Detailed Summary
This section of the chapter focuses on algebraic methods to solve pairs of linear equations. While the graphical method has its uses, it can be impractical when dealing with non-integer solutions. Therefore, we introduce two effective algebraic methods: substitution and elimination.
1. Substitution Method
The substitution method involves solving one equation for one variable and substituting that expression into the other equation. We illustrated this method with several examples, ensuring clarity through step-by-step analyses. Key steps include:
- Choose one equation and isolate one variable.
- Substitute this expression into the other equation.
- Solve for the remaining variable and substitute back to find the original variable.
Example 4 demonstrates the substitution method for the equations 7x - 15y = 2 and x + 2y = 3.
2. Elimination Method
The elimination method focuses on eliminating one variable through addition or subtraction of equations. This is often more straightforward in cases where coefficients of a variable can be made equal. In this method, one multiplies equations as necessary, then subtracts or adds to eliminate a variable. Example 8 illustrates this method through a real-world situation involving incomes and expenditures.
Summary of Concepts
- The graphical method can be impractical for non-integral solutions.
- Substitution is useful for isolating variables, while elimination simplifies the process of removal.
- Both methods can be applied to numerous scenarios involving linear equations.
By mastering these algebraic methods, students can confidently solve pairs of linear equations and apply these techniques to complex problems in mathematics.