Definition of Linear Equations
A linear equation is an algebraic equation where the highest power of the variable is 1, forming a straight line when graphed. It typically appears in the form ax + b = 0, where a and b are constants, and x is the variable. For example, 2x + 3 = 7 is a basic linear equation. The goal is to find the value of x that makes the equation true by isolating the variable.
Key Principles for Solving Linear Equations
To solve linear equations, apply inverse operations while maintaining balance on both sides. Start by simplifying using the distributive property or combining like terms, then add or subtract to move terms, and multiply or divide to isolate the variable. Remember, whatever operation is done to one side must be done to the other to preserve equality. Common tools include substitution and elimination for systems of equations.
Practical Example
Consider the equation 3x - 5 = 10. Add 5 to both sides: 3x = 15. Divide by 3: x = 5. Verify by substituting x = 5 back in: 3(5) - 5 = 10, which holds true. This step-by-step process demonstrates how linear equations model everyday scenarios, like calculating distances in travel planning.
Importance and Real-World Applications
Linear equations are foundational in algebra and essential for higher math like calculus. They model real-world situations such as budgeting, where y = 2x + 50 represents total cost (y) based on items (x). Understanding them aids problem-solving in physics, economics, and engineering, helping predict outcomes and make data-driven decisions.