Problem 58 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(-4(y+2)-5 y\) $$ (F)-9 y-8 $$ $$ (G)-9 y-6 $$ $$ (H)-9 y+2 $$ $$ (J)-9 y+8 $$

Step-by-Step Solution

Answer

(F)-9 y-8

Step 1: Distribute the terms

Distribute the \(-4\) in \(-4(y+2)\) to get \(-4y -8\), so the new equation is \(-4y -8 -5y\).

Step 2: Combine like terms

Combine like terms \(-4y\) and \(-5y\) to get \(-9y\). So the equation becomes \(-9y -8\).

Step 3: Final simplified expression

The final simplified expression is \(-9y -8\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial in simplifying algebraic expressions. Like terms are those terms in an equation that have the same variable raised to the same power. In the given exercise, we're dealing with the terms \( -4y \) and \( -5y \). Both of these terms have the variable \( y \) raised to the first power. Combining these like terms simply means to add or subtract the coefficients (the numbers in front of the variables).

For example, when you combine \( -4y \) and \( -5y \) you add the coefficients \( -4 \) and \( -5 \) to get \( -9 \) and keep the variable \( y \), resulting in \( -9y \). This process is similar to how you would combine apples with apples or oranges with oranges; you just add or subtract the quantities, but the fruit (or in our case, the variable) stays the same.

Home practice can enhance this skill. Experiment with different types of terms to become more comfortable with recognizing and combining like terms effectively.

Distributive Property

The distributive property is a fundamental concept that allows you to multiply a single term by each term within a parenthesis in an algebraic expression. This property expresses that \( a(b+c) = ab + ac \). In our exercise, the distributive property was applied to the expression \( -4(y+2) \), distributing the \( -4 \) to both \( y \) and \( +2 \) inside the parenthesis.

By applying the property, the expression becomes \( -4 \times y + (-4) \times 2 = -4y - 8 \). It's vital to pay attention to the signs, as a negative multiplied by a positive is a negative, which is why we get \( -8 \) and not \( +8 \). The distributive property is powerful because it allows us to simplify expressions before combining like terms, making the entire solving process more manageable.

Practicing with different numerical and variable factors will help to instill this property into memory. For instance, try expanding \( 3(x - 4) \) or \( -2(p + 3q) \) to get a hang of the property.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \) or \( y \)), and operators (like add, subtract, multiply, and divide). The objective in algebra is often to simplify these expressions to a point where they are easier to work with or even solve for certain variables.

An example of an algebraic expression is \( 2x + 3 \). Expressions can become quite complex, and understanding how to simplify them is an essential algebra skill. In our original problem, \( -4(y+2)-5y \) is an algebraic expression that we simplified step by step.

Remember, simplification may involve several processes, such as applying the distributive property and combining like terms as we did in the exercise. A good grasp of algebraic expressions serves as the foundation for most algebra problems, so it's very beneficial to practice simplifying a variety of expressions.