Problem 25 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x+9=18 $$

Step-by-Step Solution

Answer

The solution to the equation \(x + 9 = 18\) is \(x = 9\)

Step 1: Identifying the equation

The given equation is \(x + 9 = 18\). The aim is to find the value of the variable \(x\).

Step 2: Subtract 9 from both sides

To isolate \(x\), subtract 9 from both sides of the equation. The equation then becomes \(x + 9 - 9 = 18 - 9\)

Step 3: Simplify the equation

The equation simplifies to \(x = 18 - 9\)

Step 4: Calculating the value of x

Subtracting 9 from 18 gives 9. Therefore, the value of \(x\) is 9.

Key Concepts

Algebraic Equations

An algebraic equation is a statement that two expressions are equal, which typically contains one or more variables. In the example exercise, the equation is \(x + 9 = 18\), which expresses a relationship between the variable \(x\) and the numbers 9 and 18. To solve such equations, the main goal is to find the value of the variable that makes the equation true.

To achieve this, one must perform operations that will not change the equality, such as adding, subtracting, multiplying, or dividing the same value on both sides of the equation. This keeps the balance of the equation intact, just like a set of scales in equilibrium. It's essential, especially for beginners in algebra, to understand that whatever you do to one side of the equation, you must also do to the other side.

Isolate the Variable

Isolating the variable means rearranging the equation so that the variable you're solving for is on one side of the equation, and everything else is on the other side. To isolate the variable in the example \(x + 9 = 18\), we needed to remove the 9 from the left side. This was accomplished by subtracting 9 from both sides, as subtraction is the inverse operation of addition.

Why is isolation important? Because it gives us the value of the variable without any other numbers or variables attached to it. This step is crucial and often the focus of solving algebraic equations as it brings us closer to the solution, which in this case was identifying that \(x = 9\).

Simplifying Expressions

Simplifying expressions is an integral part of solving algebraic equations. It implies reducing the expressions to a more manageable form while keeping the value unchanged. When we subtracted 9 from both sides in \(x + 9 = 18\), we were simplifying the equation by combining like terms. The simplified form of the equation became \(x = 18 - 9\), which is then further simplified to \(x = 9\).

It's vital to understand and recognize like terms since they are the key to simplification. Like terms are terms that contain the same variables raised to the same power, such as \(2x\) and \(3x\), or \(7y^2\) and \(4y^2\). Being adept at simplifying expressions can make problems considerably easier to solve.

Basic Algebra

Basic algebra is centered around understanding variables, expressions, and equations. It lays the groundwork for more complex mathematical concepts. A firm grasp of basic algebra includes knowing how to perform operations with numbers and variables alike and understanding the properties of these operations such as commutative, associative, and distributive laws.

In the context of our example, knowing that \(18 - 9\) equals 9 is part of the computation skills required in algebra. Recognizing that addition and subtraction are inverse operations, and that one can be used to undo the other, is also vital. Basic algebra is largely about finding patterns and applying rules to manipulate equations to achieve a desired form - in most cases, to solve for a variable.