Welcome to basic algebra. In algebra, we use symbols called variables to represent unknown values. Variables are typically represented by letters like x, y, a, or b. Constants, on the other hand, are fixed numerical values like 2, negative 5, or pi. For example, in the equation x plus 2 equals 5, x is a variable whose value we need to find, while 2 and 5 are constants. By solving this equation, we can determine that x equals 3.
Algebraic expressions combine variables and constants using mathematical operations. The four basic operations are addition, subtraction, multiplication, and division. For example, 3x plus 5 is an expression that adds the constant 5 to 3 times the variable x. When working with expressions, we can combine like terms, which are terms with the same variables and exponents. For instance, 3x plus 5x equals 8x because we're adding the same variable x. On the graph, we can see how the expression 3x plus 2 creates a linear relationship, and how combining like terms works by adding the values at the same x-coordinate.
Now let's learn how to solve linear equations, which have one variable with power 1. The key principle is that whatever you do to one side of the equation, you must do to the other side to maintain equality. Let's solve the equation 3x plus 5 equals 2x minus 7. First, we move all variable terms to one side by subtracting 2x from both sides. This gives us 3x minus 2x equals negative 7 minus 5. Simplifying, we get x equals negative 12. We can verify our solution by substituting x equals negative 12 back into the original equation. 3 times negative 12 plus 5 equals negative 36 plus 5, which is negative 31. And 2 times negative 12 minus 7 equals negative 24 minus 7, which is also negative 31. Since both sides are equal, our solution is correct.
Algebra is especially useful for solving real-world problems. To solve word problems, first identify the unknown values and assign variables to them. Then write an equation based on the information given, solve it, and check if your answer makes sense. Let's look at an example: A rectangular garden has a perimeter of 36 meters. If the length is 4 meters more than the width, find the dimensions of the garden. First, we assign variables: let w be the width, and l be the length. We know that l equals w plus 4. The perimeter formula for a rectangle is 2 times width plus 2 times length equals perimeter. So our equation is 2w plus 2l equals 36. Substituting l equals w plus 4, we get 2w plus 2 times (w plus 4) equals 36. Simplifying: 2w plus 2w plus 8 equals 36, which gives us 4w plus 8 equals 36. Subtracting 8 from both sides: 4w equals 28. Dividing by 4: w equals 7 meters. Therefore, the length l equals w plus 4, which is 7 plus 4, or 11 meters. So the garden is 7 meters wide and 11 meters long.
Let's summarize what we've learned about basic algebra. First, algebra uses variables, represented by letters, to stand for unknown values. Second, algebraic expressions combine variables and constants using mathematical operations like addition, subtraction, multiplication, and division. Third, linear equations can be solved by isolating the variable through a series of steps that maintain the equality of both sides. Fourth, word problems are solved by translating real-world situations into algebraic equations. Finally, algebra provides powerful tools that allow us to model and solve a wide range of real-world problems. As you continue your mathematical journey, these fundamental algebraic concepts will serve as building blocks for more advanced topics.