Problem 34 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ (8 x-10 y+7 z)+5(3 x+2 y-4 z) $$

Step-by-Step Solution

Answer

23x - 13z

Step 1: Distribute the 5

Apply the distributive property to the term with the 5. Multiply 5 by each term inside the parentheses: $$5(3x + 2y - 4z) = 5 \times 3x + 5 \times 2y + 5 \times (-4z) = 15x + 10y - 20z$$

Step 2: Rewrite the expression

Combine the original terms with the distributed terms: $$8x - 10y + 7z + 15x + 10y - 20z$$

Step 3: Combine like terms

Group the like terms (the terms with the same variables) together and add them: $$ (8x + 15x) + (-10y + 10y) + (7z - 20z) = 23x + 0y - 13z = 23x - 13z$$

Key Concepts

Distributive Property

To simplify the given expression, we first use the distributive property. This property states that when you multiply a number by a sum of terms inside parentheses, you distribute the multiplication to each term.

For example, in the expression
$$5(3x + 2y - 4z),$$ we multiply 5 by each of 3x, 2y, and -4z.
This results in $$5 \times 3x = 15x,\ 5 \times 2y = 10y,\ 5 \times (-4z) = -20z.$$
The rewritten expression is $$15x + 10y - 20z.$$
This process helps to break down complex expressions into simpler parts and is a vital first step in algebraic simplification.

Combining Like Terms

After distributing, the next step is to combine like terms. Like terms are terms with the same variable raised to the same power. Combining them simplifies the expression further.

In the given expression after distribution:
$$8x - 10y + 7z + 15x + 10y - 20z,$$ we identify the like terms:

• 8x and 15x are like terms.
• -10y and 10y are like terms.
• 7z and -20z are like terms.

Combining 8x and 15x, we get $$8x + 15x = 23x.$$ Combining -10y and 10y, we get $$-10y + 10y = 0.$$ Combining 7z and -20z, we get $$7z - 20z = -13z.$$
Hence, the expression now is $$23x + 0 - 13z.$$
We'll simplify further in the next section.

Algebraic Simplification

After combining like terms, the final step is to simplify the expression wherever possible. Simplification makes the expression easier to understand and work with.

From the previous section, we have the expression:
$$23x + 0 - 13z.$$
We notice that the term $$0$$ can simply be omitted since adding zero does not change the value of the expression.
Therefore, we can simplify this to $$23x - 13z.$$ This is the final simplified form of the given expression.

Remember, algebraic simplification helps in solving problems more efficiently and reduces complexity. Mastery of this skill is essential for success in algebra.