Problem 122 - Introductory Algebra for College Students

Question

will help you prepare for the material covered in the next section. Remember that a solution of an equation in two variables is an ordered pair. Let \(x=0\) and find a solution of \(x+2 y=0\)

Step-by-Step Solution

Answer

The solution to the equation when \(x=0\) is the ordered pair (0,0)

Step 1: Substitute x's Value Into the Equation

Given the equation \(x+2y=0\) and we're also given that \(x=0\). Substitute \(x=0\) into the equation to get: \(0+2y=0\).

Step 2: Solve y's Value

Now simplify the equation to determine the value of \(y\). Given \(2y=0\), dividing both sides of the equation by 2 will yield the value of \(y\), i.e., \(y=0\).

Step 3: Order Pair Solution

The solution to an equation in two variables is an ordered pair (x,y). With \(x=0\) and \(y=0\), the solution is therefore the ordered pair (0,0).

Key Concepts

Ordered Pair Solutions

Understanding ordered pair solutions is a fundamental aspect of solving algebraic equations with two variables. An ordered pair is a set of numbers that represent the coordinates of a point on a Cartesian plane, usually written in the form \( (x, y) \). When dealing with equations that involve two variables, each solution consists of an x-value and a corresponding y-value that make the equation true.

As seen in the exercise, finding an ordered pair solution involves assigning a value to one variable and then solving the equation for the other variable. This process simplifies the equation and allows for the determination of the precise point at which the variables meet the conditions of the original equation. In simpler terms, for every x there is a y, and together, they form a complete solution to the equation.

Substitution Method

The substitution method is an effective technique for solving systems of equations, and it's also useful for finding an ordered pair solution to a single equation with two variables. This method involves substituting one variable with a known value or expression, making it easier to solve for the remaining variable.

In the given example, you are instructed to let \( x=0 \) and to find a solution for the equation \( x+2y=0 \). By substituting \( x \) with 0, the equation becomes much simpler, as you are essentially removing one variable from the equation. After substitution, you solve for the other variable, which, in this case, happens to result in \( y=0 \) as well. The substitution method is a cornerstone of algebra that helps in breaking down more complex problems into manageable chunks.

Algebraic Equations

At the heart of the exercise is the concept of algebraic equations. An algebraic equation is a mathematical statement indicating that two expressions are equal, often containing one or more variables. For example, in the equation \( x+2y=0 \) from the exercise, \( x \) and \( y \) are the variables, and the equation implies that when you add \( x \) to twice the value of \( y \) you get zero.

Algebraic equations can range from very simple, like the one given in the exercise, to much more complex equations involving multiple terms and variables. The main goal when solving these equations is to isolate the variable(s) to find their values. This process might involve one or several mathematical operations, such as addition, subtraction, multiplication, division, and sometimes factoring. Mastering algebraic equations is vital for anyone studying mathematics, as they form the basis for much of modern mathematics and its applications.