Problem 52 - Prealgebra

Question

Suppose \(x+y=3 .\) Find \(y\) if: $$x=-2$$

Step-by-Step Solution

Answer

We find that the value of \(y\) is 5.

Step 1: Identify Given Information

We are provided with the equation \(x + y = 3\), and we are told that \(x = -2\). Our task is to find the value of \(y\).

Step 2: Substitute Known Value of x

Replace \(x\) in the equation with the given value. This gives us:\[-2 + y = 3\]

Step 3: Solve for y

To isolate \(y\), add 2 to both sides of the equation:\[-2 + y + 2 = 3 + 2\]Which simplifies to:\[y = 5\]

Key Concepts

Variables

In algebra, variables are symbols that represent numbers whose values can change. They serve as placeholders, often noted as letters like \(x\) or \(y\), to stand in for these unknown values. Understanding variables is crucial because they allow us to formulate and solve equations by representing unknowns in mathematical relationships. In the equation \(x + y = 3\), both \(x\) and \(y\) are variables. The equation tells us that their sum equals 3. By using variables, we can easily manipulate the equation to find one variable when the other is known. In this exercise, knowing that \(x = -2\) helps us find the value of \(y\). Without variables, expressing such relationships would be cumbersome and limited to specific numbers.

Substitution

Substitution is a fundamental technique used to solve equations, involving replacing a variable with a specific value or another expression. It simplifies the equation by reducing the number of variables involved, making it easier to solve. In the given exercise, we know \(x = -2\), and we apply substitution by replacing \(x\) in the equation \(x + y = 3\) with \(-2\). This gives us the equation: \(-2 + y = 3\).

• Substitution can be used when the value of a variable is known, allowing you to solve for the remaining unknowns.
• This technique is particularly helpful in more complex problems where direct solving is not feasible.

The goal of substitution is to create a simpler equation that can then be solved using basic arithmetic operations. Substitution connects directly to solving the equation for the variable \(y\), which leads to the solution \(y = 5\).

Linear Equations

Linear equations are equations of the first degree, which means each term has an exponent of 1. They form a straight line when graphed on a coordinate plane. A typical linear equation can have two variables, like \(x\) and \(y\), and is often expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations are fundamental in algebra as they model relationships between quantities that maintain a constant rate of change.When solving linear equations, the objective is to isolate one of the variables. In the exercise with the equation \(x + y = 3\), we apply substitution and basic algebraic operations to find \(y\). Because the equation is linear, we only need simple addition and subtraction to solve it once the substitution is made. This simplicity is one of the reasons why linear equations are often the first type of equations introduced in algebra. They provide a straightforward way to demonstrate how variables and constants relate and change.