Problem 45 - Prealgebra

Question

Suppose \(x+y=5 .\) Find \(x\) if: $$y=2$$

Step-by-Step Solution

Answer

The value of \(x\) is 3.

Step 1: Understand the given equation

We are given the equation \(x + y = 5\). This means if you add \(x\) and \(y\) together, the result will be 5.

Step 2: Substitute the known value of y

We know that \(y = 2\). So replace \(y\) in the equation \(x + y = 5\) with 2: \(x + 2 = 5\).

Step 3: Solve for x

To find the value of \(x\), subtract 2 from both sides of the equation:\(x + 2 - 2 = 5 - 2\).This simplifies to \(x = 3\).

Key Concepts

Solving Equations

When you solve an equation, the goal is to find the unknown numbers or variables present in the equation. In our original exercise, the equation is given as \(x + y = 5\). This tells us that the sum of \(x\) and \(y\) is 5. To solve such equations, we apply techniques that maintain equality by performing the same operation on both sides of the equation. This keeps the equation balanced just like a balanced weighing scale.

Let’s take our specific example. We substituted \(y\) with 2, transforming the equation into \(x + 2 = 5\). To solve for \(x\), we need to get \(x\) by itself on one side of the equation. This involves removing the 2 from the left side. We do this by subtracting 2 from both sides. Mathematically, this looks like:

• First, write the equation: \(x + 2 = 5\).
• Then subtract 2 from both sides to isolate \(x\): \(x + 2 - 2 = 5 - 2\).
• Simplify to find \(x = 3\).

These steps involve basic operations and logic to arrive at the value of \(x\). This same technique can be applied to all kinds of linear equations.

Variables Substitution

Variables are symbols like \(x\), \(y\), or \(z\) that represent unknown values. Substitution is a process where we replace a variable with a known value. This is particularly handy when solving equations because it allows us to simplify the problem.

In our example, we are given that \(y = 2\). This is a direct value that can substitute \(y\) in the equation \(x + y = 5\). Once we substitute \(2\) for \(y\), our equation turns into \(x + 2 = 5\).

• This substitution eliminates one variable, making the problem easier to solve.
• It reduces the equation to a simpler form, where only one variable remains unknown.

Substitution is a fundamental algebraic technique that helps to manage equations with multiple variables.

Basic Algebra Concepts

Basic algebra involves working with numerical and literal symbols to find unknown quantities. It is the foundation for higher math. Understanding concepts such as equations, variables, and operational rules is crucial for solving algebraic problems.

Consider an equation as a statement of balance. The left-hand side equals the right-hand side. When we have an unknown, our objective is to find its value by manipulation. This involves operations like addition, subtraction, multiplication, and division.

For example, in the equation \(x + y = 5\):

• \(x\) and \(y\) are variables representing unknown values.
• Addition is used to combine variables \((x + y)\).
• To solve, you may need to isolate one variable by using basic operations.

These basic concepts ensure that, with practice, anyone can solve simpler or more complex algebraic equations.