Problem 34 - Prealgebra

Question

Solve the equation. \(-4.22 x+7.8=-6.3 x\)

Step-by-Step Solution

Answer

x ≈ -3.75

Step 1: Move Variable Terms to One Side

Start by moving all terms containing the variable x to one side of the equation. Add \(6.3x\) to both sides of the equation:\(-4.22x + 6.3x + 7.8 = -6.3x + 6.3x\)This simplifies to:\(2.08x + 7.8 = 0\)

Step 2: Isolate the Variable Term

Next, we want to isolate the term with x. Subtract 7.8 from both sides of the equation:\(2.08x + 7.8 - 7.8 = 0 - 7.8\)This simplifies to:\(2.08x = -7.8\)

Step 3: Solve for x

To find the value of x, divide both sides by 2.08:\(x = \frac{-7.8}{2.08}\)Simplify the fraction or perform the division:\(x \approx -3.75\)

Key Concepts

Understanding Algebra Basics

Algebra is a branch of mathematics that uses symbols to represent numbers and operations. When we solve an algebraic equation, the goal is to find the value of the variable that makes the equation true. This might sound complicated, but it becomes simpler as you break down each step.

The foundational concept in algebra is the equation. Think of an equation as a balance scale. The goal is to keep this scale balanced while finding the unknown value, often represented as 'x'. Algebra basics include rules for adding, subtracting, multiplying, and dividing both sides of the equation. Remember, whatever you do to one side of the equation, you must do to the other

• Start by identifying similar terms.
• Perform operations in a logical sequence.

These basic principles will help solve more complex problems as you advance in your math journey.

The Art of Isolation of Variable

Isolating the variable is crucial when solving an algebraic equation. It means getting the variable (usually represented as 'x') alone on one side of the equation, which allows us to determine its value. This process involves a series of logical steps.

In our example, the equation is \(-4.22x + 7.8 = -6.3x\). To isolate 'x', one technique is to move all terms with 'x' to one side. Here is how it's done:

• Add \(6.3x\) to both sides to cancel the \(-6.3x\) on the right.
• The equation simplifies to \(2.08x + 7.8 = 0\).

Once 'x' terms are on one side, we work to make 'x' completely alone. We do this by removing any additional numbers through subtraction or division - in this case, subtract 7.8 from both sides and then divide by 2.08. This methodical approach is the heart of isolating variables.

Navigating Fractions and Decimals

Working with fractions and decimals can be intimidating, but they are just another way to represent numbers. In linear equations, fractions and decimals often appear, and it’s important to handle them confidently.

In the example we're discussing, the number \(-4.22\) is a decimal, as are \(7.8\) and \(-6.3\). Here's how you can manage them:

• When targeting decimals, align them with similar precision for addition or subtraction.
• If multiplying or dividing, ensure each step is calculated to maintain accuracy, like using a calculator for precision.

The solution involves dividing \(-7.8\) by \(2.08\). This particular division may not yield a whole number, instead providing a decimal, \(-3.75\). Simplifying equations with fractions and decimals is a critical skill, laying the groundwork for more intricate mathematical problem-solving.