Question

Use the substitution method to solve the linear system. $$\begin{aligned} &-x+y=0\\\ &2 x+y=0 \end{aligned}$$

Short answer

So, the solution to the given system of equations is x = 0 and y = 0.

Step-by-step solution

Express 'y' in terms of 'x' from the first equation

First, rearrange the first equation \( -x + y = 0 \) to solve for 'y'. This gives \( y = x \).

Substitute 'y' into the second equation

Next, substitute \( y = x \) into the second equation \( 2x + y = 0 \). This gives \( 2x + x = 0 \) or \( 3x = 0 \).

Solve for 'x'

By solving the equation \( 3x = 0 \), obtain \( x = 0 \).

Substitute 'x' value into 'y = x'

Finally, substitute \( x = 0 \) into the equation \( y = x \), deriving \( y = 0 \).

Key concepts

Solving Linear Systems

When you're faced with simultaneous linear equations, your goal is to find the values of the unknown variables that make all the equations true at the same time. Solving linear systems is like finding the meeting point on a map where different paths cross—a point of agreement that fits all the given information.

Using algebraic methods to solve these systems is a fundamental skill in mathematics, as it applies to many fields ranging from engineering to economics. It can be visualized as finding the point of intersection between two lines on a graph. However, not all systems have a unique solution; some have infinitely many solutions, and some none at all. The method chosen to solve a system of equations depends on the structure of the system and the preference of the solver.

Substitution Technique

The substitution method is an algebraic approach for finding the exact solution to a system of equations. This technique involves isolating one variable in one of the equations and then 'substituting' this into the other equation.

Consider it a step-by-step process of replacement that reduces the system to a single equation with one variable. The steps followed generally involve isolating a variable, substituting it into another equation, and solving for the remaining variable. Once you've found that one variable, you backtrack to find the others. The process relies on the principle that if two expressions are equal to the same thing, they are equal to each other—a foundational concept in algebra known as the transitive property.

Algebraic Equations

At the heart of algebra are algebraic equations. These are mathematical statements that assert the equality of two expressions. They come in many forms, such as linear, quadratic, polynomial, and more, with the simplest being linear equations, represented by the formulae like, for example, \( ax + b = 0 \).

Variables and constants interact within an equation through operations like addition, subtraction, multiplication, and division. Solving these equations means finding the value of the variables that make the equation true. The process of solving an equation involves performing operations that will 'free' the variable, typically resulting in the variable being isolated on one side of the equation.

Systems of Equations

A system of equations is a set of two or more equations that have variables in common. The objective when solving such a system is to find values for each of the variables that will satisfy all equations in the system at once. Systems can be classified based on the number of solutions they possess—if they have no solutions, they are inconsistent; if they have exactly one solution, they are consistent and independent; and if they have an infinite number of solutions, they are consistent and dependent.

There are a few different methods to solve these systems, such as graphing, substitution, and elimination. Each approach has its benefits, and often, the choice of method depends on the particular system you're dealing with. For example, graphing is useful for visualizing the solutions, while substitution and elimination are more algebraic and precise methods.