An equation is a mathematical statement that shows two expressions are equal, separated by an equals sign. Think of an equation like a balance scale - both sides must be equal for the equation to be true. For example, 3 plus 2 equals 5, or x plus 4 equals 10. In equations, we have variables like x or y that represent unknown values, coefficients which are numbers that multiply variables, constants which are fixed numbers, and solutions which are the values that make the equation true.
Equations can be classified by their degree, which is the highest power of the variable. Linear equations have degree 1 and form straight lines when graphed, like 2x plus 3 equals 7. Quadratic equations have degree 2 and form parabolas, such as x squared minus 5x plus 6 equals 0. Polynomial equations have degree 3 or higher and create more complex curves. The degree tells us important information about the equation's behavior and how many solutions it might have.
To solve linear equations, we use inverse operations to isolate the variable while maintaining equality on both sides. Think of it like a balance scale - whatever we do to one side, we must do to the other. Let's solve 2x plus 5 equals 13. First, we subtract 5 from both sides to get 2x equals 8. Then we divide both sides by 2 to get x equals 4. We can check our answer by substituting 4 back into the original equation: 2 times 4 plus 5 equals 13, which is correct.
Multi-step linear equations require a systematic approach. When variables appear on both sides, like 3x plus 7 equals 2x plus 12, we first move all variables to one side by subtracting 2x from both sides, giving us x plus 7 equals 12. Then we subtract 7 from both sides to get x equals 5. For equations with parentheses like 2 times the quantity x plus 3 equals 14, we first distribute to get 2x plus 6 equals 14, then subtract 6 from both sides to get 2x equals 8, and finally divide by 2 to get x equals 4. Remember to avoid common mistakes like forgetting to distribute or making sign errors when moving terms.
Quadratic equations have the form ax squared plus bx plus c equals zero, where a is not zero. These equations can be solved using factoring or the quadratic formula. Let's solve x squared minus 5x plus 6 equals zero by factoring. We look for two numbers that multiply to 6 and add to negative 5, which are negative 2 and negative 3. So we factor as x minus 2 times x minus 3 equals zero. Using the zero product property, either x minus 2 equals zero or x minus 3 equals zero, giving us x equals 2 or x equals 3. The quadratic formula provides a universal method for any quadratic equation and is especially useful when factoring is difficult.