Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(3(x+2)=x+30\)

Step-by-Step Solution

Answer

The solution to the given equation is \(x = 12\).

Step 1: Distributive Property

Applying the distributive property of multiplication over addition, multiply 3 with each term within the parentheses. This gives \(3*x + 3*2 = x + 30\), which simplifies to \(3x + 6 = x + 30\).

Step 2: Collect Like Terms

Collect like terms by subtracting 'x' from both sides and subtracting '6' from both sides. This gives \(3x - x = 30 - 6\), which simplifies to \(2x = 24\).

Step 3: Solve for x

Solve for 'x' by dividing each side of the equation by 2. This gives \(x = 24/2\), which simplifies further to \(x = 12\).

Step 4: Check the Solution

Substitute 'x' with '12' in the original equation \(3(x+2)=x+30\). This gives \(3(12+2) = 12 + 30\), simplifying to \(3*14 = 42\), finally which simplifies to \(42 = 42\). Since both sides of the equation are equal, the solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing a factor across terms inside parentheses.
For the given equation \(3(x+2) = x + 30\), applying the distributive property means multiplying each term inside the parentheses by 3.
This results in \(3 \cdot x + 3 \cdot 2\), leading to the expression \(3x + 6\).

• This step helps in breaking down complex expressions into simpler parts.
• It's essential for efficiently simplifying and solving equations.
• The main idea is to "distribute" the multiplier to each term inside the brackets.

Once you've applied the distributive property, you can more easily manipulate the equation to isolate terms and solve for the variable.

Like Terms

In algebra, combining like terms is a crucial step to simplify expressions and solve equations.
Like terms are terms in an expression that have the exact same variable raised to the same power.
In the equation \(3x + 6 = x + 30\), like terms can be identified and combined:

• The terms \(3x\) and \(x\) are like terms because they both contain the variable \(x\).
• Subtract \(x\) from both sides to get \(3x - x = 2x\).
• The constants \(6\) and \(30\) can also be combined by subtracting \(6\) from \(30\), simplifying to \(24\).

Combining these like terms simplifies the equation to \(2x = 24\), a much easier problem to solve.
This process is essential for reducing the problem to a form where the solution becomes apparent.

Checking Solutions

After solving an equation, it's always important to check if the solution is correct.
This is done by substituting the found value back into the original equation to ensure both sides equal.
For the exercise, after finding \(x = 12\), we substitute back into the original equation \(3(x+2) = x + 30\):

• Calculate \(3(12+2)\) which simplifies to \(3 \cdot 14 = 42\).
• On the right, substitute \(x = 12\) to get \(12 + 30 = 42\).

Both sides equal 42, confirming that our solution \(x = 12\) is indeed correct.
Checking your answer in this manner helps to ensure accuracy, guaranteeing that your steps were correct and the solution is valid.