A linear equation in one variable is a fundamental concept in algebra. It's called 'one variable' because it contains only one unknown quantity, typically represented by letters like x, y, or z. The equation has the general form a-x plus b equals zero, where 'a' is the coefficient that cannot be zero, 'b' is the constant term, and 'x' is the variable we need to find. For example, 2x plus 5 equals 11 is a linear equation in one variable.
The fundamental principle for solving linear equations is maintaining balance, just like a scale. Whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side. For example, to solve x plus 3 equals 7, we subtract 3 from both sides. This gives us x plus 3 minus 3 equals 7 minus 3, which simplifies to x equals 4. The equation remains balanced throughout the process.
Let's solve a linear equation step by step. Consider the equation 2x minus 1 equals 5. Step 1: Isolate the variable term by adding 1 to both sides. This gives us 2x minus 1 plus 1 equals 5 plus 1, which simplifies to 2x equals 6. Step 2: Isolate the variable by dividing both sides by 2. This gives us 2x divided by 2 equals 6 divided by 2, which simplifies to x equals 3. We can check our answer by substituting x equals 3 back into the original equation: 2 times 3 minus 1 equals 6 minus 1 equals 5, which is correct.
Linear equations can be visualized on a number line, which helps us understand the solution geometrically. For our equation 2x minus 1 equals 5, we found that x equals 3. On the number line, we can mark this solution as a point at position 3. This visual representation shows that the solution is the specific value on the number line where the equation is satisfied. The number line helps us see that there is exactly one solution to a linear equation in one variable.
Linear equations in one variable are fundamental tools in mathematics with many real-world applications. Let's summarize the key points: they involve one variable with a power of 1, follow the form a-x plus b equals zero, require maintaining balance when solving, and have exactly one solution. For example, if a store sells books for 15 dollars each and the total cost is 75 dollars, we can find how many books were bought using the equation 15x equals 75. Solving this gives us x equals 5 books. Linear equations are essential problem-solving tools in mathematics, science, and everyday life.