Problem 50 - Prealgebra

Question

Suppose \(x+y=5 .\) Find \(x\) if: $$y=3$$

Step-by-Step Solution

Answer

\(x = 2\).

Step 1: Identify the Given Information

We are given the equation \(x+y=5\) and the value of \(y\) is provided as \(y=3\).

Step 2: Substitute the Value of \(y\) into the Equation

Replace \(y\) in the equation \(x+y=5\) with the given value of \(y=3\). This gives us the equation: \(x + 3 = 5\).

Step 3: Solve for \(x\)

To find the value of \(x\), we need to isolate \(x\) on one side of the equation. Subtract 3 from both sides: \(x = 5 - 3\). This simplifies to \(x = 2\).

Step 4: Verify the Solution

Substitute \(x = 2\) and \(y = 3\) back into the original equation \(x + y = 5\) to ensure it satisfies the equation. \(2 + 3 = 5\), which is correct.

Key Concepts

Substitution Method

The substitution method is a powerful tool in algebra that allows us to solve systems of equations by replacing one variable with its known value. It's like solving a puzzle by figuring out which piece fits where. In our exercise, we started with the equation:

• \(x + y = 5\)

We were given \(y = 3\). This means we can substitute 3 in place of \(y\) in the original equation. So our equation becomes:

• \(x + 3 = 5\)

Substituting makes equations easier to solve because it reduces the number of unknowns. Imagine the equation is a balancing scale. By substituting \(y\)'s value, all we need to do next is balance the scale to find \(x\).' It's a straightforward method, especially great for beginners to grasp and apply in more complex problems later on.

Basic Algebra

Basic algebra involves using operations like addition, subtraction, multiplication, and division to solve equations. Once we've substituted \(y\) with 3 in the equation \(x + 3 = 5\), our job is to isolate \(x\). This is done by removing the 3 that's added to \(x\).

• Subtract 3 from both sides: \(x + 3 - 3 = 5 - 3\)

The left side simplifies to \(x\) because the \(+ 3\) and \(- 3\) cancel each other out. The right side calculates to 2, so:

• \(x = 2\)

This step embodies the essence of basic algebra: manipulate equations to get the unknown by itself. Applying arithmetic operations symmetrically keeps the equation balanced, leading us to the solution.

Verifying Solutions

Verifying solutions is a crucial final step to ensure that our answer is correct. It's like checking your work to boost confidence in the solution. After finding \(x = 2\), we substitute both \(x\) and \(y\) back into the original equation to verify:

• Original equation: \(x + y = 5\)

Substitute \(x = 2\) and \(y = 3\):

• \(2 + 3 = 5\)

Since both sides of the equation are equal, our solution is verified. Verifying ensures accuracy, and in algebra, it's a habit worth forming. This step reassures us that even if a mistake was made early on, it'd be caught during the problem-checking phase.