Algebra Basics: Simplifying Expressions & Formulas

In algebra, we work with two types of numbers: variables and constants.Variables are letters that can represent different numbers. We often use x, y, and z.Constants are fixed numbers that don't change their value.A single variable can represent many different values. Let's see how x can be equal to different numbers.When we combine variables and constants, we create algebraic expressions.Let's see what happens when we substitute different values into an expression.Let's look at a real-world example: the distance formula.To combine like terms, we need terms with the same variables raised to the same powers.Like terms have exactly the same variables with the same exponents.Unlike terms have different variables or different powers, and cannot be combined.Let's practice combining like terms. Here's our first example: four y plus two y minus y.We can combine these terms because they all have the same variable y. Four y plus two y is six y, minus y equals five y.Here's another example with squared terms: three x squared plus two x squared plus x squared.Let's look at some common mistakes to avoid when combining terms.For example, two x plus three y does NOT equal five x y. These are unlike terms and cannot be combined.The distributive property tells us that when we multiply a number by a sum, we can multiply that number by each term separately.Let's break this down step by step. Two times x plus three means we multiply two by x, and two by three.The same principle applies when we have negative numbers. Let's look at three times two x minus four.When distributing with multiple terms, we multiply each term inside the parentheses by the number outside.Now let's look at some practice problems. Try these on your own, remembering to distribute the number to each term inside the parentheses.Let's check the solutions. Five x plus ten, six x minus two, and negative three x minus twelve.When we have a negative sign in front of parentheses, we need to be careful about how we remove them.The key rule is that when we remove parentheses preceded by a negative sign, ALL terms inside change their signs.Let's look at another example. In negative three x minus four, both terms will change signs when we remove the parentheses.Let's break this down step by step.Now let's look at a more complex example. When we have five minus two x plus three in parentheses.The negative sign affects both terms inside the parentheses. The positive two x becomes negative two x, and the positive three becomes negative three.Here are some practice problems for you to try. Remember to change ALL signs when removing parentheses preceded by a negative.Let's solve this complex expression by breaking it down into steps we've learned.First, we'll use the distributive property on both sets of parentheses.Next, we'll write our expression with all terms in sequence.Now we can combine like terms. We have six x minus x, which gives us five x. And eight plus five gives us thirteen.Let's verify our answer by substituting a value for x.Let's use x equals 2 to check both the original and simplified expressions.Before we finish, let's review some common mistakes to avoid.Let's review the key points for solving complex algebraic expressions.Thanks for learning algebra with Spark.E!