Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 3[2 x-(x+7)]=5(x-3) $$

Step-by-Step Solution

Answer

The solution for given equation is \( x = -3 \).

Step 1: Simplify initial equation

Distribute the multiplier in the equation. This gives us: \( 3[2x -(x+7)] = 5(x - 3) \) simplifies to \( 3x - 21 = 5x - 15 \).

Step 2: Re-arrange the terms

Move all x terms to one side of the equation, resulting in \( 3x - 5x = -15 + 21 \), which simplifies to \( -2x = 6 \).

Step 3: Solve for variable 'x'

Isolate x by dividing both sides of the equation by -2. This yields \( x = -3 \).

Step 4: Verify solution

Substitute x = -3 back into the original equation to verify. It results in \( 3[2*(-3) - ((-3)+7)] = 5((-3) - 3) \). Simplifying confirms that both sides equals \( -18 \), therefore x = -3 is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that helps us simplify equations by distributing a multiplication over addition or subtraction. In the original exercise, we start with the equation:

• \[ 3[2x -(x+7)] = 5(x - 3) \]

Using the distributive property, we expand both sides:

• Left side: Multiply 3 by each term inside the brackets, resulting in \[ 3 \times 2x - 3 \times (x + 7) \]. This simplifies to \[ 6x - 3x - 21 \].
• Right side: Similarly, multiply 5 by each term inside the parentheses, leading to \[ 5x - 15 \].

This process helps in breaking down expressions and making them easier to solve. By distributing correctly, we arrive at a simplified, linear equation: \[ 3x - 21 = 5x - 15 \], ready for the next step in solving.

Equation Verification

Verification is about checking whether our solution satisfies the original equation. After isolating and solving for the variable, it's crucial to substitute back to confirm accuracy. In the provided solution, we found \( x = -3 \).

• Substitute \( x = -3 \) into the original equation: \[ 3[2(-3) - ((-3) + 7)] = 5((-3) - 3) \]
• Simplify both sides. For the left side: \[ 3[-6 - 4] = 3(-10) = -30 \].
• For the right side: \[ 5(-6) = -30 \].

Both sides equal \( -30 \), confirming our solution \( x = -3 \) is correct. Verification affirms all work and shows that no mistakes were made, such as incorrect distribution or arithmetic errors.

Isolating Variables

Isolating the variable is a crucial part of solving equations, allowing us to find the unknown value. In our equation, we start from the simplified form:

• \[ 3x - 21 = 5x - 15 \]

To isolate \( x \), follow these steps:

• Rearrange terms to get all \( x \) terms on one side: \[ 3x - 5x = -15 + 21 \].
• This simplifies to \[ -2x = 6 \].
• Finally, divide both sides by \( -2 \) to solve for \( x \): \[ x = \frac{6}{-2} = -3 \].

Rearranging and isolating the variable streamlines the equation-solving process, ensuring that we accurately determine the variable's value. It's a step-by-step strategy to simplify while maintaining balance in the equation.