Problem 45 Simplify each expression. $$ ... [FREE SOLUTION] (2024)

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To simplify algebraic expressions, we often need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression -5y + 3 - 1 + 5 + y - 7, the terms -5y and y are like terms because they both contain the variable y. Similarly, 3, -1, and -7 are like terms because they are constants. Combining like terms makes an expression simpler and easier to work with.

• Identify the terms with the same variable or constants.
• Add or subtract the coefficients of like terms.
• Rewrite the expression using the simplified terms.

In our example, we first identify and combine the constant terms: 3, -1, and -7. Adding these together, we get 3 - 1 - 7 = -5.
Then, we combine the variable terms -5y and y. When combined, we get -5y + y = -4y.
This process helps us simplify the original expression to -4y - 5.

In algebraic expressions, constants are numbers without any variables. They remain the same no matter what value the variable takes. Constants are important because they can be combined to simplify expressions.
Consider the expression -5y + 3 - 1 + 5 + y - 7.

• Here, 3, -1, and 5 are constants.
• We combine these values by simple addition and subtraction.
• When we add these constants: 3 - 1 = 2 and 2 - 7 = -5.

This simplification reduces the constants to a single term, making the entire expression simpler. It is a crucial step in solving algebraic problems efficiently.

Variables are symbols like x, y, z, and so on, that represent numbers. In algebraic expressions, variables allow us to create equations that can be solved for different values. Each variable term has a coefficient, which is the number multiplied by the variable.
For example, in -5y + 3 - 1 + 5 + y - 7:

• The variables are y.
• The coefficients of y are -5 and 1.

To combine like terms with variables, we simply add or subtract their coefficients. Here, combining -5y and y (which is 1y) involves adding the coefficients: -5 + 1 = -4. Therefore, the variable terms simplify to -4y.
Understanding how to work with variables is essential because they form the basis of algebraic equations and functions.