Problem 18 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ -2(x+5)=30 $$

Step-by-Step Solution

Answer

The solution is \( x = -20 \) and the answer is verified.

Step 1: Distribute the Negative Coefficient

Distribute the \(-2\) on the left-hand side of the equation to both terms inside the parentheses. This gives you:\[-2 \cdot x - 2 \cdot 5 = 30\]So, the equation becomes:\[-2x - 10 = 30\]

Step 2: Isolate the Variable Term

To isolate the term with the variable \(x\), add 10 to both sides of the equation:\[-2x - 10 + 10 = 30 + 10\]This simplifies to:\[-2x = 40\]

Step 3: Solve for x

Now, divide both sides by \-2\ to solve for \(x\):\[x = \frac{40}{-2}\]Simplifying gives:\[x = -20\]

Step 4: Verify the Solution

Substitute \(x = -20\) back into the original equation to verify the solution:\[-2(-20 + 5) = 30\]Simplify within the parentheses:\[-2(-15) = 30\]Calculate the left side:\[30 = 30\]Both sides match, confirming the solution is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept in algebra that makes solving equations easier by simplifying expressions. It involves multiplying a single term by each term inside a set of parentheses. In this exercise, we work with the equation: \(-2(x+5)=30\). Here, the distributive property allows us to eliminate the parentheses by distributing the \(-2\) to both \(x\) and \(5\).

• The multiplication of \(-2\) by \(x\) results in \(-2x\).
• The multiplication of \(-2\) by \(5\) results in \(-10\).

After applying the distributive property, the equation becomes \(-2x - 10 = 30\). The goal is to simplify the equation step by step so that solving for the variable becomes straightforward.

Isolating Variables

Isolating the variable is another essential step when solving linear equations. Once the equation is simplified using the distributive property, the next task is to get the variable on its own on one side of the equation.Here, we start with the equation \(-2x - 10 = 30\). To isolate \(x\), we aim to have it stand on its own, which involves removing any additional numbers from its side of the equation. By adding 10 to both sides, we effectively neutralize the -10, as demonstrated below:

• Add 10 to both sides: \(-2x - 10 + 10 = 30 + 10\).
• The equation then becomes: \(-2x = 40\).

Finally, to solve for \(x\), we divide everything by \(-2\). This operation completely isolates \(x\) and provides us with the solution \(x = -20\). The focus here is on step-by-step manipulation to simplify and isolate.

Checking Solutions

Checking your solution is like double-checking your homework—it ensures correctness and builds confidence in the solution you found. Once we calculated that \(x = -20\), we need to verify that this value satisfies the initial equation. To check our work, we'll substitute \(x\) back into the original equation. Here's how we proceed:

• Substitute \(x = -20\) into \(-2(x + 5) = 30\).
• Simplify within the parentheses: \(-2(-20 + 5)\).
• This becomes \(-2(-15)\).
• Finally, simplify: \(30 = 30\).

Both sides of the original equation are equal, confirming our solution is correct. This process of verifying is crucial as it validates that the steps were executed accurately and the solution is indeed valid.