Problem 54 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \(y\) is 0.

Step 1: Start with the Given Equation

We are provided with the equation \(x + y = 3\). This is the equation we will work with to find the value of \(y\).

Step 2: Substitute the Known Value of x

We know from the problem that \(x = 3\). Substitute this value into the equation \(x + y = 3\) to get \(3 + y = 3\).

Step 3: Isolate y

To find \(y\), we need to isolate it on one side of the equation. We will subtract 3 from both sides of the equation: \(3 + y - 3 = 3 - 3\).

Step 4: Simplify the Equation

Simplify both sides of the equation to yield \(y = 0\).

Key Concepts

Equation Solving

Solving equations is a fundamental part of algebra that involves finding the value of variables that make an equation true. When we are presented with an equation, it's a mathematical statement indicating that two expressions are equal. Our task is to determine the unknown values that will satisfy this equality. In the context of basic algebra, the unknown values are typically represented by variables, like \(x\) or \(y\).

Here's the process most commonly used for solving basic algebraic equations:

• Identify the given equation and understand the values and terms involved.
• Use basic arithmetic operations to simplify the equation, if needed.
• Perform operations to isolate the variable of interest on one side of the equation.
• Solve for the variable to find its value.

In our original exercise, the equation \(x + y = 3\) becomes our starting point. To solve for any one variable in such an equation, we need to decide on methods that will efficiently lead us to the solution.

Substitution Method

The substitution method is a well-known technique used to solve systems of equations. It's especially helpful when one of the variables is already isolated or can be isolated easily on both sides of the equations. The idea is to substitute a known value or expression for a variable, which simplifies the equation.

To use the substitution method, follow these steps:

• Identify which variable can be easily replaced with a given value or another expression.
• Substitute the given value or expression into the original equation in place of the identified variable.
• Simplify the equation as necessary and solve for the remaining variable.

In our exercise, since we're given \(x = 3\), we substitute 3 directly into the equation \(x + y = 3\), resulting in \(3 + y = 3\). This substitution sets the stage for quickly solving for \(y\), without needing to process convoluted calculations.

Isolating Variables

Isolating variables is the process of manipulating an equation to get a specific variable on one side by itself. This is a crucial step when solving equations because it allows us to find the exact value of the unknown variables.

Here's how you can isolate a variable:

• Look at the equation and identify the variable you need to solve for.
• Use inverse operations—addition, subtraction, multiplication, or division—to move other terms to the opposite side of the equation.
• Simplify the equation to have the variable alone on one side.

In our example, after substituting \(x = 3\) into the equation \(x + y = 3\), we proceed to isolate \(y\). By subtracting 3 on both sides (\(3 + y - 3 = 3 - 3\)), we effectively eliminate the constant term from one side, simplifying the equation to \(y = 0\). This isolating step is critical, as it allows us to pinpoint the exact numeric value of the variable based solely on the operations allowed in the equation.