Problem 19 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ 2 x+6(2 x+3)=-10 $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Expand the Equation

Start by distributing the 6 across the terms inside the parentheses. The equation is: \[ 2x + 6(2x + 3) = -10 \]After expanding, this becomes: \[ 2x + 12x + 18 = -10 \]

Step 2: Combine Like Terms

Combine the terms that have \( x \) in them. We have \( 2x + 12x \), so:\[ 14x + 18 = -10 \]

Step 3: Isolate the Variable

Subtract 18 from both sides to move the constant term to the right side of the equation:\[ 14x = -10 - 18 \]Simplifying gives:\[ 14x = -28 \]

Step 4: Solve for x

Divide both sides by 14 to solve for \( x \):\[ x = \frac{-28}{14} \]This simplifies to:\[ x = -2 \]

Step 5: Check Your Solution

Substitute \( x = -2 \) back into the original equation to verify the solution. The original equation is: \[ 2(-2) + 6(2(-2) + 3) = -10 \]Calculate inside the parentheses first:\[ 2(-2) + 6(-4 + 3) = -10 \]This becomes:\[ -4 + 6(-1) = -10 \]Finally, simplify:\[ -4 - 6 = -10 \]This is true, so the solution is verified.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions. It allows you to multiply a single term by each term inside parentheses in an expression. In our original equation:\[2x + 6(2x + 3) = -10\]we use the distributive property to expand the expression inside the parentheses. This means we multiply 6 by both \(2x\) and 3, which gives us:

• 6 times \(2x\) equals \(12x\)
• 6 times 3 equals 18

So, after applying the distributive property, our equation transforms to:\[2x + 12x + 18 = -10\]This expansion simplifies the equation, making it easier to solve in the following steps.

Combining Like Terms

Once we have expanded the equation using the distributive property, the next step is to simplify it by combining like terms. Like terms are terms that have the same variable component. In our equation, the like terms are those with the variable \(x\):\[2x + 12x + 18 = -10\]To combine the like terms \(2x\) and \(12x\), simply add their coefficients together:

• \(2x + 12x = 14x\)

After combining like terms, the equation simplifies to:\[14x + 18 = -10\]This step reduces the equation to a simpler form that is much easier to solve.

Isolation of Variables

After combining like terms, the goal is to isolate the variable, which means getting the variable \(x\) alone on one side of the equation. Our equation now looks like this:\[14x + 18 = -10\]To isolate \(x\), we need to move the constant 18 to the other side. This is done by subtracting 18 from both sides:\[14x = -10 - 18\]Simplifying the right side gives:\[14x = -28\]Now, to solve for \(x\), divide both sides by 14:\[x = \frac{-28}{14}\]This further simplifies to:\[x = -2\]By isolating the variable, we've determined the solution for \(x\).

Solution Verification

Verifying the solution is an essential final step in solving equations to ensure that your answer is correct. Here, we take the solution \(x = -2\) and substitute it back into the original equation to see if it satisfies the equation:\[2(-2) + 6(2(-2) + 3) = -10\]First, calculate inside the parentheses:

• \(2(-2) = -4\)
• \(2(-2) + 3 = -4 + 3 = -1\)
• \(6(-1) = -6\)

Now, add the results:\[-4 - 6 = -10\]Since both sides of the equation equal \(-10\), our solution \(x = -2\) is verified, confirming its correctness. Verifying ensures that no mistakes were made along the way and reassures you about the accuracy of your solution.