Problem 51 - Elementary Algebra

Question

Solve. $$ 3.1(2 x-3)+0.5=22.2 $$

Step-by-Step Solution

Answer

The value of \( x \) is 5.

Step 1: Distribute the Constant

Start by distributing the constant 3.1 across the terms inside the parentheses: \[ 3.1(2x - 3) = 3.1 \times 2x - 3.1 \times 3 \]This simplifies to: \[ 6.2x - 9.3 \]

Step 2: Simplify the Equation

Substitute the distributed terms back into the equation:\[ 6.2x - 9.3 + 0.5 = 22.2 \]Combine like terms:\[ 6.2x - 8.8 = 22.2 \]

Step 3: Isolate the Variable Term

Add 8.8 to both sides of the equation to isolate the term with the variable:\[ 6.2x = 22.2 + 8.8 \]This simplifies to:\[ 6.2x = 31 \]

Step 4: Solve for the Variable

Divide both sides of the equation by 6.2 to solve for \( x \):\[ x = \frac{31}{6.2} \]Perform the division to find:\[ x = 5 \]

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions and solve equations. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. For example, if you have an expression like \( a(b + c) \), you can use the distributive property to transform it into \( ab + ac \). This is exactly what you need to do in the original exercise when dealing with \( 3.1(2x - 3) \). By applying the distributive property:

• Multiply 3.1 by each term inside the parentheses.
• This turns \( 3.1(2x - 3) \) into \( 6.2x - 9.3 \).

By understanding and applying the distributive property, you can simplify complex expressions and make equations easier to work with.

Mastering Variable Isolation

Variable isolation is crucial when solving equations because it means getting the variable by itself on one side of the equation. This allows you to find out what the variable equals. In the context of the original problem, after simplifying using the distributive property, you have the equation \( 6.2x - 8.8 = 22.2 \). The goal is to isolate \( x \).Here’s how you can achieve variable isolation:

• First, you need to eliminate any constants from the variable side, so add 8.8 to both sides to get \( 6.2x = 31 \).
• Next, divide each term by the coefficient of the variable, which is 6.2, to isolate \( x \).

After these steps, you will have \( x = 5 \). This shows that variable isolation helps in systematically narrowing down towards the solution.

Simplifying Equations

Equation simplification combines the process of reducing complexity in an equation to make it straightforward to work with. This typically involves distributing constants (as you did with the distributive property), combining like terms, and performing arithmetic operations. In the provided exercise,

• After distribution, replace the expression with \( 6.2x - 9.3 + 0.5 = 22.2 \).
• Next, simplify further by combining \( -9.3 + 0.5 \) which equals \( -8.8 \).
• This simplifies the equation to \( 6.2x - 8.8 = 22.2 \).

Simplifying equations is a valuable skill, as each step clearly moves you towards isolating the variable and solving the problem. It reduces the chance of errors and helps manage more complex equations easily.