Understanding the Goal of Solving Linear Equations
Solving a linear equation involves finding the value(s) of the variable(s) that make the equation true. A linear equation is an algebraic equation where each term has an exponent of 1, and when graphed, it forms a straight line. The primary goal is to isolate the unknown variable (often 'x') on one side of the equality sign (=) by performing inverse operations to both sides of the equation.
Key Principles and Step-by-Step Method
The fundamental principle for solving linear equations is to maintain balance: whatever operation is applied to one side of the equation must also be applied to the other side. The general steps include: 1. Simplify both sides of the equation (distribute, combine like terms). 2. Use addition or subtraction to move all terms containing the variable to one side and all constant terms to the other side. 3. Use multiplication or division to isolate the variable, typically making its coefficient 1. Always perform operations in the reverse order of operations (PEMDAS/BODMAS).
Practical Example: Solving 3x - 7 = 8
Let's solve the equation `3x - 7 = 8`. First, to isolate the term `3x`, add `7` to both sides: `3x - 7 + 7 = 8 + 7`, which simplifies to `3x = 15`. Next, to isolate `x`, divide both sides by `3`: `3x / 3 = 15 / 3`. This yields `x = 5`. To verify, substitute `x = 5` back into the original equation: `3(5) - 7 = 15 - 7 = 8`. Since `8 = 8`, the solution is correct.
Importance and Applications of Linear Equations
Linear equations are foundational to mathematics and have wide-ranging applications in various fields. They are used to model relationships between quantities in science, engineering, economics, and everyday situations. For instance, calculating distances, determining costs, predicting outcomes, or understanding rates of change often involves setting up and solving linear equations. Mastery of this concept is essential for tackling more complex algebraic topics and real-world problem-solving.