Problem 24 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Six times the sum of a number and eight is 30 . Find the number.

Step-by-Step Solution

Answer

The number is -3.

Step 1: Understand the problem

The problem states that six times the sum of a number and eight equals 30. We need to translate this into an equation.

Step 2: Define the variable

Let the unknown number be represented by the variable \( x \).

Step 3: Translate the words into an equation

According to the problem, six times the sum of a number and eight can be written as \( 6(x + 8) \). This is equal to 30, so the equation is: \[ 6(x + 8) = 30 \]

Step 4: Solve for the variable

First, distribute the 6 on the left side: \[ 6x + 48 = 30 \] Next, subtract 48 from both sides to isolate the term with the variable: \[ 6x = 30 - 48 \] So we get: \[ 6x = -18 \]

Step 5: Solve for x

To find \( x \), divide both sides by 6: \[ x = \frac{-18}{6} \] So, \[ x = -3 \]

Step 6: Verify the solution

Substitute \( x = -3 \) back into the original equation to check: \[ 6(-3 + 8) = 30 \] which simplifies to \[ 6(5) = 30 \] This is true, so \( x = -3 \) is correct.

Key Concepts

Equation Solving

When solving algebra word problems, the first step is to translate the problem into an equation. The example problem states that six times the sum of a number and eight equals 30.
Translation into an equation involves these steps:

• Identify the unknown number, which we'll call x.
• Sum this number with eight, giving us (x + 8).
• Multiply this sum by six, forming 6(x + 8).
• Set this equal to 30, resulting in the equation 6(x + 8) = 30.

The equation is our starting point for further steps to find the unknown number.

Variable Isolation

Isolating the variable means getting the unknown number by itself on one side of the equation. Here's how to do it:
First, we need to simplify the equation. We achieve this by applying the distributive property to remove the parenthesis:
[ 6(x + 8) = 30 ] becomes [ 6x + 48 = 30 ].
Next, we need to move all constant terms to the opposite side of the equation from the variable term.
Since we have 48 on the same side as 6x, we subtract 48 from both sides to isolate the term involving x:
[ 6x + 48 - 48 = 30 - 48 ] simplifies to [ 6x = -18 ]. Now, we'll move on to the next crucial step of finding the exact value of x.

Distributive Property

The distributive property is a useful tool in solving equations and is expressed as
[ a(b + c) = ab + ac ]. In our example, we have
[ 6(x + 8) = 30 ]. We use the distributive property to multiply each term inside the parentheses by 6. As a result,
[ 6x + 48 = 30 ]. This step is critical because it helps eliminate the parenthesis, making it easier to isolate and solve for the variable in subsequent steps.

Verification of Solution

After finding the variable value, it's important to verify the solution. Substituting the found value back into the original equation checks its correctness.
In our case, we found that x = -3. Substitution into the original equation gives
[ 6(-3 + 8) ]. Simplifying the inside of the parentheses gives
[ 6(5) = 30 ], and
[ 30 = 30 ] confirms our solution is valid. This verification step ensures accuracy and helps reinforce your understanding of the problem-solving process.