Problem 133 - Prealgebra and Introductory Algebra

Question

Is \(-3\) a solution of the equation \(x-7=-10 ?\)

Step-by-Step Solution

Answer

Yes, \(-3\) is a solution to the equation

Step 1: Substitution

Substitute \(x\) in equation \(x-7=-10\) with \(-3\). So we get \(-3-7\).

Step 2: Solving the equation

To process the ─ operation we get to \(-10\).

Step 3: Comparison

As \(-10 = -10\), \(-3\) is a valid solution for the provided equation

Key Concepts

Solving Equations

Solving equations is a fundamental skill in prealgebra that involves finding the value of a variable that makes an equation true. When solving equations, the goal is to isolate the variable on one side of the equation, often using inverse operations to both sides equally. For example, if we start with an equation like \(x - 7 = -10\), we need to manipulate it to identify what value for \(x\) satisfies the equation.
In the example above, by solving the equation for \(x\), you can confirm whether a particular number, such as \(-3\), is indeed a solution. It's important to follow the principle of performing the same operation on both sides of the equation to maintain the equality.

• Ensure you simplify each side as much as possible.
• Double-check your solution by substituting it back into the original equation.

This methodical process helps solidify understanding and accuracy when dealing with various equations.

Substitution Method

The substitution method is a valuable technique in algebra used to verify if a certain value is a solution to an equation. This involves replacing the variable with the given number and checking if the equation remains balanced. For our specific problem, we use the substitution method to test if \(-3\) is a solution for \(x - 7 = -10\).

By substituting \(x\) with \(-3\), the equation becomes \(-3 - 7\). Subsequently evaluating this, we find that it simplifies to \(-10\).

• If the left side of the equation equals the right side after substitution, the value is a solution.
• This method confirms or debunks the proposed solution through direct calculation.

Using substitution is an essential approach, especially useful when equations are complex and it provides confirmation through direct evidence of solving.

Integer Operations

Understanding integer operations is crucial when solving equations involving integers, as it forms the basis for efficient algebraic manipulation. Integers include positive and negative whole numbers. The operations typically involve addition, subtraction, multiplication, or division. In this case, the focus is on the subtraction of integers.
When you perform operations with integers like \(-3 - 7\), it is essential to understand the rules:

• Subtracting a number is the same as adding its negative.
• Adding or subtracting negative numbers requires careful attention to their signs.

For \(-3 - 7\), we are effectively moving 7 steps to the left on a number line starting from \(-3\), leading to \(-10\). Grasping these operations not only helps solve equations correctly but also enhances skills in numerical accuracy and logical thinking.