Problem 10 - Prealgebra

Question

Solve the equation. \(-1.33 x-45.22=0\)

Step-by-Step Solution

Answer

The solution is \(x \approx -34.02\).

Step 1: Add 45.22 to both sides

To isolate the term with the variable, add 45.22 to both sides of the equation. This gives us: \(-1.33x - 45.22 + 45.22 = 0 + 45.22\). Simplifying this gives:\(-1.33x = 45.22\).

Step 2: Divide both sides by -1.33

Now, divide each side of the equation by -1.33 to solve for \(x\): \[x = \frac{45.22}{-1.33}\].

Step 3: Calculate the result

Perform the division to find the value of \(x\). Calculate \(\frac{45.22}{-1.33}\), which gives:\(x \approx -34.02\).

Key Concepts

Solving Equations

Solving equations is a fundamental process in algebra that involves finding the value of the variable that makes an equation true. An equation is essentially a statement that two expressions are equal. To solve an equation like the one given \(-1.33x - 45.22 = 0\), we take steps to simplify and manipulate the equation until we have isolated the variable on one side. This allows us to determine its value.

The equation-solving process often involves:

• Adding or subtracting terms to both sides to simplify or rearrange the equation.
• Multiplying or dividing both sides by a number to further isolate the variable.
• Performing basic arithmetic operations accurately to ensure the correct result.

Mastery in solving equations is crucial because it builds the foundation for tackling more complex problems in algebra and other math-related fields. Always keep a balance in equations: whatever you do to one side, you must do to the other.

Isolation of Variable

Isolation of a variable is a critical step in solving equations. It involves using algebraic operations to rearrange an equation so one side is just the variable you want to solve for. In the given problem, \(-1.33x - 45.22 = 0\), we aim to get \(x\) by itself on one side of the equation.

To achieve isolation:

• Identify the term containing the variable, in this case, \(-1.33x\).
• Eliminate any additional terms on the same side as the variable, such as by adding \(45.22\) to both sides of the equation. This helps to neutralize the constant, leading to \(-1.33x = 45.22\).
• Adjust the coefficient of the variable to \(1\) by dividing both sides by \(-1.33\).

Isolation simplifies the problem to a single operation that directly provides the value of the variable. This concept applies across various types of algebraic equations, helping make seemingly complex calculations manageable.

Division in Equations

Division is a key operation used to solve equations, especially when dealing with a coefficient multiplying the variable. When solving our equation, once you've added \(45.22\) to both sides and simplified it to \(-1.33x = 45.22\), division is necessary to isolate \(x\).

Here's how division aids in solving equations:

• Neutralize the coefficient: Divide every term in the equation by the coefficient of the variable. This converts the equation to a form where the variable stand-alone. In our exercise, we divide by \(-1.33\) to get \(x = \frac{45.22}{-1.33}\).
• Ensure an accurate calculation: Division between numbers results in a precise value for the variable, here calculated as \(x \approx -34.02\).
• Maintain equation balance: Just like other operations, division must be applied equally on both sides of the equation to uphold the equation's truth.

Understanding division is crucial when handling equations because it frequently appears as a step in multiple algebraic methods and scenarios. Correct execution of division translates algebraic manipulations into accurate mathematical conclusions.