Problem 14 - Prealgebra

Question

Solve each equation. $$0.8+10 y=-0.7$$

Step-by-Step Solution

Answer

The solution is \(y = -0.15\).

Step 1: Isolate the Variable Term

We start by moving the constant term to one side of the equation to isolate the variable term. The equation is \(0.8 + 10y = -0.7\). Subtract 0.8 from both sides to get:\[10y = -0.7 - 0.8\]

Step 2: Simplify the Equation

On the right side, perform the subtraction:\[10y = -1.5\]

Step 3: Solve for the Variable

Now, divide both sides of the equation by the coefficient of \(y\), which is 10, to solve for \(y\):\[y = \frac{-1.5}{10}\]

Step 4: Simplify the Result

Simplify \(\frac{-1.5}{10}\) by performing the division:\[y = -0.15\]

Key Concepts

Solving Equations

Solving linear equations involves finding the value of the variable that makes the equation true. The key idea is to isolate the variable on one side of the equation to determine its value. In the given exercise, the equation is \(0.8 + 10y = -0.7\). Here, our goal is to solve for \(y\). To achieve this:

• We begin by moving the constant term to the other side of the equation. This is done through the operation of subtraction, allowing us to keep the equation balanced.
• After subtracting 0.8 from both sides, the equation becomes \(10y = -1.5\).
• The next step is dividing both sides of the equation by 10, the coefficient of \(y\), to find the value of \(y\).

By systematically performing these operations, we determine that \(y = -0.15\). Mastering these steps helps in solving more complex equations effectively.

Prealgebra

Prealgebra lays the foundational skills necessary for understanding and solving basic mathematical problems. It forms the backbone of algebra by covering arithmetic and introducing simple algebraic reasoning. In the context of this exercise, prealgebra involves:

• Understanding variables, which are symbols that represent unknown numbers, such as \(y\) in our equation.
• Recognizing the importance of keeping equations balanced, meaning whatever operation you perform on one side, you must do the same on the other side.
• Knowing how to rearrange and combine terms to simplify the expression and isolate variables.

These skills are crucial as a stepping stone towards more complex algebraic concepts and prepare students for handling more elaborate mathematical challenges.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves making an equation or expression easier to work with, without changing its value. In this exercise, simplifying is a key process:

• After subtracting 0.8 from both sides of the equation \(0.8 + 10y = -0.7\), we simplify the right-hand side to \(-1.5\).
• This simplification allows us to have the variable term \(10y\) on one side, making it easier to solve for \(y\).
• Finally, dividing \(-1.5\) by 10 simplifies the expression to \(-0.15\), which is the solution for \(y\).

Consistently applying these simplification techniques ensures clarity and accuracy in working through algebraic expressions and equations, paving the way for solving them efficiently.