Problem 11 - Prealgebra

Question

Solve the equation. \(-12.5 x+13.5=0\)

Step-by-Step Solution

Answer

The solution is \(x = 1.08\).

Step 1: Understand the Equation

The equation given is a linear equation: \(-12.5 x + 13.5 = 0\). Our goal is to find the value of \(x\) that makes this equation true.

Step 2: Isolate the x-term

To solve for \(x\), we need to isolate the term involving \(x\). Start by subtracting 13.5 from both sides of the equation to get: \(-12.5x = -13.5\).

Step 3: Solve for x

Now, divide both sides by \(-12.5\) to solve for \(x\): \(x = \frac{-13.5}{-12.5}\).

Step 4: Simplify the Solution

Divide \(-13.5\) by \(-12.5\) to find the value of \(x\). The negatives cancel each other out, so \(x = \frac{13.5}{12.5}\). When you calculate this division, you get \(x = 1.08\).

Key Concepts

Solving Equations

Solving equations is like unlocking puzzles. You have to figure out what value of a variable, like \( x \), makes an entire equation true. A linear equation, such as \(-12.5x + 13.5 = 0\), involves terms that are either constants or the variable raised to the first power. To solve such equations, follow these steps:

• Identify the equation you need to solve. Each term in your equation needs to be understood.
• Work systematically by performing the same operations on both sides of the equation to maintain equality.

An equation is solved when the variable is isolated, and you have a number on the other side of the equal sign showing its value. Remember, the operations you choose should gradually reveal the value of the variable.

Isolating Variables

Isolating variables is a cornerstone of solving equations. Here, the aim is to get the variable (often \( x \)) by itself on one side of the equation. Imagine peeling layers off an onion until all that's left is the core variable you seek. For the given equation \(-12.5x + 13.5 = 0\):

• Start by removing the constant on the same side as the variable. In this case, subtract 13.5 from both sides to strip away the constant. You then have \(-12.5x = -13.5\).
• Next, to isolate \( x \), eliminate the coefficient of the variable by dividing both sides by \(-12.5\). This operation provides \( x = \frac{-13.5}{-12.5} \).

If each step is followed correctly, isolation becomes effective, giving you a simple form of \( x \) that can now be easily calculated.

Simplifying Fractions

Simplifying fractions help in calculations by reducing them to their simplest form. This means breaking down the fraction so that the numerator and denominator are as small as possible, but the same ratio is preserved.In the step solution, after isolating the variable, you end up with \( x = \frac{-13.5}{-12.5} \). Here's what to do next:

• Notice that negative signs on both the numerator and the denominator cancel out since a negative divided by a negative yields a positive result.
• At this point, the fraction simplifies to \( \frac{13.5}{12.5} \).
• You can divide 13.5 by 12.5 using a calculator or long division. When you do this division, you find that \( \frac{13.5}{12.5} \approx 1.08 \).

Being adept at simplifying fractions not only makes your answers cleaner and more concise but also builds a strong foundation for tackling more complex math problems.