Problem 16 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$5(x-1)+7(x+4)$$

Step-by-Step Solution

Answer

The simplified expression is \(12x + 23\).

Step 1: Distribute the 5 across the first parentheses

To simplify the expression, first distribute the 5 into the terms inside the parentheses. This means you multiply 5 by each term inside: \[5(x-1) = 5 \cdot x - 5 \cdot 1 = 5x - 5\]

Step 2: Distribute the 7 across the second parentheses

Next, distribute the 7 into the terms inside the second set of parentheses. Multiply 7 by each term inside:\[7(x+4) = 7 \cdot x + 7 \cdot 4 = 7x + 28\]

Step 3: Combine the expanded expressions

Write out the expression with the newly distributed terms from steps 1 and 2:\[5x - 5 + 7x + 28\]

Step 4: Combine like terms

Now, combine the like terms in the expression. The like terms are the terms involving x and the constant terms:\[(5x + 7x) = 12x\]\[(-5 + 28) = 23\]So the simplified expression is:\[12x + 23\]

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that helps us simplify expressions without altering their value. It allows you to "distribute" a single term over terms inside a set of parentheses. This is done by multiplying the single term by each term in the parentheses.
Let's think about it this way: if you have a term outside of parentheses, like 5 in the expression \(5(x-1)\), you must multiply it by each term inside the parentheses. Here's how it works:

• Multiply 5 by x, which gives you \(5x\).
• Then, multiply 5 by -1, resulting in -5.

Thus, the expression becomes \(5x - 5\). This technique not only looks cool, it's very useful in simplifying complex algebraic expressions. Once you've distributed all applicable terms, you're ready to move on to simplifying further.

Combining Like Terms

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In our example, after applying the distributive property, we have:

• The expression: \(5x - 5 + 7x + 28\)
• Notice there are two terms with an x: \(5x\) and \(7x\).
• By adding \(5x\) and \(7x\), we get \(12x\).

Similarly, notice there are two constant terms: -5 and 28.
If we’re adding those, we simply do the arithmetic: \(-5 + 28 = 23\).
These are often referred to as "like terms," meaning they can be combined because they share the same characteristics. This step helps boil down complex expressions to more manageable forms by gathering similar items together.

Simplification Steps

The simplification process in algebra involves several key techniques. By following these steps, we can efficiently solve any algebraic expression:1. **Apply Distributive Property:** - As mentioned earlier, it's crucial to distribute any constants or variables across parentheses first. 2. **Rewrite the Expression:** - After distribution, rewrite the expression to show all terms clearly, like we did when we got \(5x - 5 + 7x + 28\). 3. **Identify Like Terms:** - Look for terms with similar variables or constant numbers. 4. **Combine Like Terms:** - Add or subtract the like terms (e.g., combine \(5x\) and \(7x\) to get \(12x\)).5. **Final Simplification:** - Ensure there are no further like terms to combine. The expression should be in its simplest form, which in our case was \(12x + 23\).By executing these steps systematically, you simplify expressions confidently. This foundational skill is immensely beneficial in algebra, making complex equations easier to solve.