Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$1.5 x-0.7=0.4(3-5 x)$$

Step-by-Step Solution

Answer

x = \(\frac{19}{35}\)

Step 1: Distribute the Right-Hand Side

We begin by distributing the 0.4 to both terms inside the parentheses on the right side of the equation. This gives us \(1.5x - 0.7 = 0.4 \times 3 - 0.4 \times 5x\). Simplifying inside, we have \(1.5x - 0.7 = 1.2 - 2x\).

Step 2: Move All Terms Involving x to One Side

In order to solve for \(x\), we need to get all the terms with \(x\) on one side. Add \(2x\) to both sides of the equation: \(1.5x + 2x - 0.7 = 1.2\). This simplifies to \(3.5x - 0.7 = 1.2\).

Step 3: Move Constant Terms to the Other Side

To isolate the term with \(x\), add \(0.7\) to both sides of the equation: \(3.5x = 1.2 + 0.7\). Simplifying the right-hand side, we have \(3.5x = 1.9\).

Step 4: Solve for x

Now, isolate \(x\) by dividing both sides of the equation by 3.5: \(x = \frac{1.9}{3.5}\). Simplifying the fraction, we get \(x = \frac{19}{35}\).

Step 5: Simplify the Solution

The fraction \(\frac{19}{35}\) is already in its simplest form, so the solution to the equation is \(x = \frac{19}{35}\).

Key Concepts

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and is fundamental in solving equations. When solving an algebra problem like the given equation, our goal is to find the value of the unknown variable, often represented by letters like \(x\).

• Equations are expressions set equal to each other.
• Variables serve as placeholders for numbers.
• Solving an equation means finding the value of the variable that makes the equation true.

In the exercise, we are dealing with a linear equation, which involves constants and a single variable raised to the power of one. The processes used here focus on modifying the equation to maintain equality while simplifying it to reveal the variable's value.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form, making it easier to solve. The goal is to combine like terms and perform basic arithmetic to tidy up the expression. By reducing the complexity of an equation, you can focus on the core components that need solving.
In the given problem, equation simplification plays a vital role when we rearrange and combine terms. By distributing numbers through parentheses, we simplify both sides of the equation. Later, by consolidating like terms, such as combining coefficients of \( x \), we simplify further steps:

• Perform arithmetic operations like addition or subtraction.
• Combine like terms: terms that have the same variable raised to the same power.
• Adjust the equation to make isolating the variable straightforward.

Throughout this process, maintaining the balance of the equation—i.e., what you do to one side, you must do to the other—is critical to ensuring the solution remains valid.

Distributive Property

The distributive property is a key algebraic principle that allows you to remove parentheses by distributing a factor over the terms inside. In mathematical terms, it states:
\[ a(b + c) = ab + ac \]
This property is frequently used to simplify expressions and solve equations as it ensures every term inside the parentheses is multiplied by the factor outside.
In the exercise, the distributive property is used to simplify the right side of the equation: \(0.4(3 - 5x)\). By applying this property, you end up with:

• \(0.4 \times 3 = 1.2\) simplifies to \(1.2\)
• \(0.4 \times (-5x) = -2x\) simplifies to \(-2x\)

After applying the distributive property, the equation becomes easier to work with, paving the way for further simplification and isolation of the variable.

Isolation of Variable

Isolation of the variable is the step where you rearrange the equation to get the variable by itself on one side. This is the ultimate goal in solving equations, as it directly provides the value of the variable.
Using basic operations, you shift constants and terms to isolate the variable:

• Move terms with the variable to one side using addition or subtraction.
• Move constant terms to the opposite side.
• Divide or multiply the entire equation to get the variable alone.
  

In the exercise, after simplification, you add \(2x\) to both sides first, then eventually eliminate other terms by addition or subtraction until only \(3.5x = 1.9\) remains. Finally, dividing both sides by the coefficient \(3.5\) fully isolates \(x\), leading to a solution of \(x = \frac{19}{35}\). This methodical approach ensures clarity and accuracy in finding the correct variable value.