Problem 45 - Elementary Algebra

Question

Solve. $$ 1=10-3 x $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Isolate the Variable Term

First, we need to isolate the term with the variable on one side of the equation. Starting with \(1 = 10 - 3x\), we subtract 10 from both sides to get \(1 - 10 = -3x\), which simplifies to \(-9 = -3x\).

Step 2: Solve for the Variable

Now, we solve for \(x\) by dividing both sides of the equation by \(-3\). So, \(x = \frac{-9}{-3}\). This simplifies to \(x = 3\).

Key Concepts

Solving Equations

Solving equations is a central component of algebra. It involves finding the value of a variable that makes the equation true. In this particular exercise, we have a linear equation: \(1 = 10 - 3x\). To solve it, we need to isolate the variable term. We begin by understanding that the goal is to find the value of \(x\) that satisfies the equation. This process requires using basic arithmetic operations to manipulate the equation without changing its equality. The ability to solve equations efficiently relies on good foundational skills in arithmetic and comprehension of the properties of equality.
By working through this process, we ensure that the solution maintains the equality. This makes solving equations both a systematic and logical process.

Variable Isolation

Variable isolation is a key step in the process of solving equations. It refers to rearranging the equation so that the variable appears on one side. In our equation \(1 = 10 - 3x\), we aim to isolate \(x\) by putting it on one side of the equation.

• Start by subtracting \(10\) from both sides: \(1 - 10 = -3x\), leading to \(-9 = -3x\).

Once you have isolated the variable term \(-3x\), it becomes easier to solve for the actual value of the variable. Isolating variables requires understanding that whatever operation you perform on one side to manipulate the variable, you must also perform on the other side to maintain the balance of the equation.
This concept is pivotal because it allows you to focus on a singular part of the equation, facilitating more straightforward solutions.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to simplify and solve equations. In this exercise, the key to solving \(1 = 10 - 3x\) was careful manipulation of the terms involved.

• First, subtract \(10\) from both sides to simplify to \(-9 = -3x\).
• Next, divide both sides by \(-3\) to solve for \(x\), giving you: \(x = \frac{-9}{-3} = 3\).

This step-by-step manipulation shows how algebraic principles allow us to solve for unknown variables. Understanding the rules of operations like addition, subtraction, multiplication, and division, helps in executing algebraic manipulations correctly.
These productive manipulations lead us efficiently to the solution, illustrating the power of algebra in problems solving. By grasping these techniques, students can tackle a wide array of equations effectively.