Question

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Short answer

The solution to the given equation \( 3x - 5 = 10 \) is \( x = 5 \).

Step-by-step solution

Add 5 to both sides of the equation.

By adding 5 to both sides of the equation, we can remove the constant term (-5) from the left side of the equation. This results in: \[ 3x - 5 + 5 = 10 + 5 \]

Simplify the equation.

After adding 5 to both sides of the equation, we can simplify the equation as follows: \[ 3x = 15 \]

Divide both sides of the equation by 3.

To isolate x, we need to remove the coefficient (3) on the left side of the equation. To do this, we divide both sides of the equation by 3: \[ \frac{3x}{3} = \frac{15}{3} \]

Simplify the equation and find the value of x.

After dividing both sides of the equation by 3, we simplify the equation to find the value of x: \[ x = 5 \] So the solution to the given equation \( 3x - 5 = 10 \) is \( x = 5 \) .

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with using symbols and letters to represent numbers and quantities in formulas and equations. It allows us to model real-world problems and abstract concepts mathematically. The ability to work with variables gives us a lot of flexibility.
When we use letters such as \( x \) or \( y \), these symbols become placeholders that can take on different values. In the given exercise, for example, the variable \( x \) represents a number that, once determined, makes the equation true.
Algebra is useful in various fields such as science, engineering, and technology because it allows us to solve equations and make predictions about unknown quantities.

• Key components include variables, coefficients, and constants.
• Expressions are combinations of variables, numbers, and operations.
• Equations set two expressions equal to each other and include an equality sign \((=)\).

Solving Linear Equations

Solving linear equations involves finding the value of a variable that makes the equation true. Linear equations in one variable, like the one in the exercise \(3x - 5 = 10\), have a single solution.
The goal is to isolate the variable, \(x\), on one side of the equation. This process requires a systematic series of steps that use inverse operations to undo the operations applied to the variable.
The method often involves:

• Adding or subtracting terms on both sides to remove constants from the side of the variable. Example: Adding 5 to both sides: \(3x - 5 + 5 = 10 + 5\).
• Simplifying the equation to make it more manageable. Example: Simply the previous step to get: \(3x = 15\).
• Dividing or multiplying to isolate the variable and find its specific value. Example: Divide both sides by 3: \(\frac{3x}{3} = \frac{15}{3}\).

Each step applies a basic arithmetic operation that progressively makes the equation simpler. This logical sequence guarantees that the original equation remains balanced, ultimately leading to the discovery of the solution, \(x = 5\).

Mathematical Operations in Equation Solving

Mathematical operations are fundamental tasks of arithmetic — addition, subtraction, multiplication, and division — that are used to manipulate equations. These operations allow us to change the appearance of equations without altering their equality, enabling us to solve for unknown variables effectively.
In the exercise, the operations were used specifically to simplify and solve the equation \(3x - 5 = 10\):

• Addition: To counteract subtraction on the variable's side, we added 5 to both sides to eliminate \(-5\), leading to \(3x = 15\).
• Division: By dividing both sides by 3, we counteracted multiplication, resulting in \(x = 5\).

Understanding how to appropriately apply these operations is crucial as they enable one to transition between different forms of an equation while keeping it balanced. Each operation has an inverse — addition with subtraction, multiplication with division — which makes it possible to "reverse" or undo operations in solving equations. This is a key technique in algebraic problem-solving that simplifies complex equations.