Problem 47 - Prealgebra

Question

Suppose \(x+y=5 .\) Find \(x\) if: $$y=0$$

Step-by-Step Solution

Answer

When \( y = 0 \), \( x = 5 \).

Step 1: Understand the Equation

The given equation is a simple linear equation: \( x + y = 5 \). This means for any value of \( y \), \( x \) plus that value should equal 5.

Step 2: Substitute the Value of y

We are given \( y = 0 \). Substitute this value into the equation: \( x + 0 = 5 \).

Step 3: Simplify the Equation

Since \( 0 \) has no effect when added to \( x \), the equation simplifies to \( x = 5 \).

Step 4: Interpret the Result

Thus, when \( y = 0 \), \( x \) must be 5 to satisfy the original equation.

Key Concepts

Solving Linear Equations

Linear equations are equations involving two variables that create a straight line when graphed. In its simplest form, a linear equation looks like: \( ax + by = c \). The solution to a linear equation is the set of values for the variables that make the equation true.
To solve a linear equation, we need to isolate the desired variable on one side of the equation. In the given equation \( x + y = 5 \), we want to find the value of \( x \) given a specific value of \( y \). This involves a few straightforward steps:

• Substitute the known value of one variable.
• Then, solve the resulting equation for the remaining variable.

This approach ensures that the equation remains balanced, meaning that whatever you do to one side of the equation, you must do to the other. This is key in maintaining the equality of both sides.

Substitution Method

The substitution method is a frequently used technique to solve linear equations. It involves replacing one variable with its known value or another expression that represents it.
In the given exercise, we know \( y = 0 \) and we use this method to find the value of \( x \). We substitute \( y = 0 \) into the equation \( x + y = 5 \) to create \( x + 0 = 5 \). This simplification helps us solve for \( x \).
Substitution is particularly useful in systems of equations where you might substitute the expression representing the variable from one equation into another. This can transform a system of two equations into a single-variable equation, making it easier to solve.

• Replace the variable with its known value.
• Solve the simplified equation.

This method highlights how understanding relationships between variables can simplify your work.

Simplifying Expressions

Simplifying expressions helps make equations easier to solve and understand. For instance, in our step-by-step solution, after substituting \( y = 0 \), we simplify \( x + 0 \) to \( x \).

• Identify and eliminate terms that have no impact on the equation, like adding zero.
• Combine like terms to make the equation clearer.

Simplification is crucial because simpler equations are easier to solve. Always look for opportunities to reduce complexity by removing or combining terms. This can often make it clear at-a-glance what the solution should be. With practice, simplifying expressions becomes easier and can greatly speed up solving more complex equations.