Problem 95 - Algebra A Combined Function

Question

Solve. 0.07 x-5.06=-4.92

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

The given equation is \(0.07x - 5.06 = -4.92\). We need to isolate \(x\) to solve the equation.

Step 2: Move Constants to the Right Side

Add 5.06 to both sides of the equation to move the constant term to the right side. This gives us: \[0.07x - 5.06 + 5.06 = -4.92 + 5.06\] simplifying to \[0.07x = 0.14\]

Step 3: Isolate x

Divide both sides of the equation by 0.07 to solve for \(x\). Thus, we have: \[x = \frac{0.14}{0.07}\]

Key Concepts

Solving Equations

In algebra, solving equations involves finding the value of the variable that makes the equation true. An equation usually presents a balance between two expressions, which could either be numbers, variables, or a mix of both. In the given exercise, we start with the equation \(0.07x - 5.06 = -4.92\), and our goal is to determine the value of \(x\). Successfully solving an equation means identifying the precise value or values that satisfy the equation.

To solve equations efficiently, follow these general steps:

• Identify the variable you need to solve for.
• Perform operations to isolate the variable on one side of the equation.
• Simplify the equation where possible.

By using these basic steps, you can tackle linear equations systematically.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions in an equation. This is key to solving equations as it helps in isolating the variable you are interested in. For the equation \(0.07x - 5.06 = -4.92\), algebraic manipulation first helps us to rearrange terms by adding, subtracting, or performing other operations on both sides of the equation.

Here, we start by moving the constant term \(-5.06\) to the other side of the equation by adding its opposite, \(5.06\), to both sides. This eliminates the constant term from the left side, simplifying our equation to \(0.07x = 0.14\).

Steps to perform algebraic manipulation effectively include:

• Use inverse operations to move terms across the equation.
• Combine like terms to simplify expressions.
• Be consistent in applying operations on both sides of the equation.

This approach ensures the equation remains balanced as you work towards isolating the variable.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. This involves isolating the variable on one side of the equation to determine its value. It essentially means re-arranging the equation such that the variable \(x\) stands alone on one side of the equation, while everything else is on the other side.

After performing the algebraic manipulation procedure in our original problem \(0.07x = 0.14\), we reached a point where \(x\) is coupled with the coefficient \(0.07\). By dividing both sides by \(0.07\), we achieve the isolation, giving us \(x = \frac{0.14}{0.07}\), which simplifies to \(x = 2\).

To effectively isolate variables:

• Use division or multiplication, whichever applies, to eliminate coefficients.
• Ensure that the isolated variable is by itself to solve the equation.
• Perform checks by substituting back the value to see if the original equation is satisfied.

By following these steps, you can systematically isolate the variable and solve the equation with ease.