Problem 81 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 10=x-5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 15\).

Step 1: Identifying the Problem

The problem consists of a simple linear equation \(10 = x - 5\) in one variable 'x'. The task is to isolate 'x' to solve the equation.

Step 2: Applying Addition Property

To isolate 'x', add 5 to both sides of the equation to balance it. Therefore, the equation becomes: \(10 + 5 = x - 5 + 5\).

Step 3: Simplifying

Simplify both sides of the equation to solve for 'x'. The equation now simplifies to \(15 = x\). Therefore, 'x' equals 15.

Key Concepts

Isolate the Variable

Isolating the variable is often the first step toward solving a linear equation. In essence, the goal here is to manipulate the equation in a way that leaves the variable of interest, often 'x,' all by itself on one side of the equation. This is achieved usually by using inverse operations – adding where there's subtraction, multiplying where there's division – until the variable stands alone.

For instance, in the equation given in the exercise, we have the variable 'x' that we want to find, but it is not by itself. To isolate 'x,' you can add or subtract quantities to both sides, respecting the need to always keep the equation balanced. That is, whatever operation you do to one side, you must do to the other as well. By doing this, the variable 'x' will end up being isolated, making it easier to determine its value.

Addition Property of Equality

The addition property of equality is a fundamental rule in algebra. It states that you can add the same number to both sides of an equation without changing the equation's solution. This property is based on the idea that an equation signifies two expressions that are equal. If you increase both expressions by the same amount, they will remain equal.

The exercise at hand beautifully illustrates this principle. When given the equation \(10 = x - 5\), you can add 5 to both sides, which looks like this: \(10 + 5 = x - 5 + 5\). It doesn’t alter the inherent 'balance' of the equation but allows us to inch closer to finding out what 'x' is by removing the -5 that's attached to the 'x'.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves combining like terms and reducing expressions to their most basic form. In simple terms, to 'simplify' an expression means to make it as straightforward as possible. This could mean getting rid of parentheses, combining similar terms, and eliminating complex fractions.

In the step-by-step solution provided, after applying the addition property of equality, we reach the point where the equation looks like \(15 = x\). This is a significantly simpler expression than what we started with, as it directly gives us the value of 'x'. In this case, no further simplification is needed, because 'x' is already isolated and we have found our solution. Simplifying is all about making sure that equations are as easy to read and solve as possible.