Problem 30 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$16+4[5 x-4(x+2)]=7-2 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1/14\).

Step 1: Simplification

Rewrite the equation by expanding the brackets to simplify the computations. \(16+4[5x - 4(x+2)] = 7-2x\). Simplify this to \(16+20x -8x - 8=7 - 2x\). Combine like terms to simplify further to \(12x + 8 = 7 - 2x \).

Step 2: Isolate the variable x

To isolate x, we need to get all terms with x on one side. Do this by adding \(2x\) to both sides and subtracting \(8\) to both sides, we get \(14x = -1\).

Step 3: Solve for x

To solve for x, divide both sides by 14 because that is the coefficient of x. Therefore, \(x = -1/14\).

Key Concepts

Equation Simplification

When solving algebraic equations, the first step is usually to simplify the equation. This means reducing the equation into a form that is easier to work with. Look at the provided equation:

• \(16+4[5x - 4(x+2)] = 7-2x\)

Using simplification, the equation becomes clearer and more manageable. Steps to simplify include:

• Expanding brackets using the distributive property, which helps eliminate parentheses.
• Simplifying constants and coefficients by performing appropriate arithmetic operations on them.

For instance, expanding the equation gives \(16 + 20x - 8x - 8 = 7 - 2x\). By simplifying the components of the equation, you are left with terms that can further be addressed by combining them, as we will see next.

Combining Like Terms

Once the equation is simplified, the next logical step is to combine like terms. Like terms are terms in the equation that contain the same variable raised to the same power. Let's revisit the simplified equation:

• \(16 + 20x - 8x - 8 = 7 - 2x\)

To combine like terms, you need to:

• Identify all constant terms and sum them up.
• Identify variables that share the same exponent and combine their coefficients.

Following these rules on our equation, combined like terms give you:

• \(12x + 8 = 7 - 2x\)

This brings the equation to a stage where isolating the variable becomes more streamlined.

Variable Isolation

Isolating the variable means getting the variable on one side of the equation so that you can solve for it. From the equation we now have:

• \(12x + 8 = 7 - 2x\)

To isolate the variable \(x\), both sides of the equation need to be manipulated:

• Add \(2x\) to both sides to transfer the \(x\) terms to one side: \(12x + 2x + 8 = 7\).
• Subtract \(8\) from both sides to bring the constants to the opposite side: \(14x = -1\).

This step focuses on "balancing" the equation by performing the same operations on both sides. This preserves the equality while simplifying the path towards calculating the value of \(x\).

Distributive Property

The distributive property is a core concept in algebra that allows you to simplify more complex expressions. It involves multiplying a single term by each term within a bracket:

• The general formula is \(a(b + c) = ab + ac\).

Applying it to the initial equation \(4[5x - 4(x+2)]\):

• First, distribute \(4\) into each term inside the brackets.
• This results in: \(4 \cdot 5x - 4 \cdot 4(x+2)\).
• Which simplifies step by step to: \(20x - 16(x+2)\).
• Further expanding \(-16(x+2)\) gives \(-16x - 32\).
• Thus leading the equation to \(16 + 20x - 16x - 32\).

By applying the distributive property, complex expressions are broken down into more manageable parts, facilitating equation simplification and the eventual solving.