Problem 74 - Algebra and Trigonometry

Question

\(71-78\) Find the solution of the equation correct to two decimals. $$ 3.95-x=2.32 x+2.00 $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 0.59\).

Step 1: Set Up the Equation

The given equation is \(3.95 - x = 2.32x + 2.00\). We need to solve for \(x\).

Step 2: Move all terms involving x to one side

Subtract \(2.32x\) from both sides of the equation to get all \(x\) terms on one side. This gives us \(3.95 - x - 2.32x = 2.00\).

Step 3: Combine x terms

Combine the like terms involving \(x\) on the left side: \(-1x - 2.32x = -3.32x\). So, the equation simplifies to \(3.95 - 3.32x = 2.00\).

Step 4: Isolate the x term

Subtract 3.95 from both sides to get \(-3.32x = 2.00 - 3.95\). Simplifying the right side gives \(-3.32x = -1.95\).

Step 5: Solve for x

Divide both sides by \(-3.32\) to solve for \(x\): \[x = \frac{-1.95}{-3.32}\]. Simplifying this gives \(x \approx 0.59\), rounded to two decimal places.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and are equations where the highest power of the variable is one.
They can be incorporated into many real-life scenarios and are foundational to understand.The general form of a linear equation is:

• In one variable: \(ax + b = 0\)
• In two variables: \(ax + by = c\)

Here, \(a\), \(b\), and \(c\) are constants, \(x\) and \(y\) are variables. A linear equation is like a straight line in its graphical representation.In the exercise above, the equation \(3.95-x=2.32x+2.00\) is a linear equation. The goal is to find the value of \(x\) that satisfies the equation.
By performing algebraic manipulations, we isolate the variable \(x\) on one side of the equation.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. These symbols can represent numbers and are used to express mathematical relationships and operations.
It's like a language of its own, where each variable and operation follows precise rules.In the context of solving an equation such as \(3.95 - x = 2.32x + 2.00\), algebra involves:

• Identifying like terms
• Combining similar terms together
• Arranging terms to isolate the variable

By subtracting \(2.32x\) from both sides, we bring all terms with \(x\) together. Then, combining the terms \(-x - 2.32x\) simplifies to \(-3.32x\).
The purpose is to clear the equation of unnecessary elements to clearly see what \(x\) equals.

Problem Solving

Problem solving in math involves several key strategies to tackle equations effectively. Solving linear equations can seem complex at first, but breaking it down into small steps can simplify the process.For example, with the equation \(3.95 - x = 2.32x + 2.00\), consider these strategies:

• Understand the problem: Recognize the type of equation and variables involved.
• Plan a solution path: Figure out the sequence of operations needed to isolate the variable.
• Execute the plan methodically: Follow each step closely, as shown in the solution, from moving terms, combining like terms, to isolating \(x\).

Through these steps, you systematically solve for \(x\), obtaining \(x \approx 0.59\), rounded to two decimal places.
The goal is to make the process easier and more intuitive, leading to a correct solution.