Question

Solve the linear system. $$ \begin{aligned} &x-y=-4\\\ &2 y+x=5 \end{aligned} $$

Short answer

The solution to the system is \( x = -1 \), \( y = 3 \)

Step-by-step solution

Rearrange the First Equation

Rearrange the first equation to solve for \( x \) by adding \( y \) to both sides. This gives \( x = y - 4 \)

Substitute \( x \) in Second Equation

Substitute \( x \) by \( y - 4 \) in the second equation. This gives \( 2y + y - 4 = 5 \)

Solve for \( y \)

Combine like terms and then solve for \( y \) to get \( y = 3 \)

Substitute \( y \) Value into First Equation

Substitute \( y = 3 \) into first equation which is \( x = y - 4 \) to get \( x = -1 \)

Key concepts

Substitution Method

In algebra, the substitution method is a way to solve systems of equations where one equation is solved for one variable, which is then substituted into the other equations. The process reduces the number of variables, thus simplifying the problem.

For instance, with the given system:
\begin{align*}&x - y = -4 \ &2y + x = 5d{align*}The first equation is rearranged to isolate one variable (in this case, \( x \)), resulting in \( x = y - 4 \). This expression for \( x \) is then used to replace the \( x \) in the second equation, reducing the two-variable equation to a single variable equation that can easily be solved. This substitution allows us to find the value of one variable and then use it to find the value of the other.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. Each linear equation in a two-variable system, like \( x \) and \( y \), usually takes on the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

These equations describe a relationship in which the variables can change, but they maintain a constant ratio that defines their linear relationship. When solving systems containing linear equations, the goal is to find values for the variables that make both equations true simultaneously.

Algebraic Manipulation

Algebraic manipulation involves rewriting equations or expressions to simplify them or to make them more useful for a particular purpose, such as solving for a variable. This can include a variety of techniques such as expanding brackets, factoring, combining like terms, and moving terms from one side of an equation to the other by addition or subtraction.

For example, in our problem:
\begin{align*}&2y + (y - 4) = 5d{align*}we combine like terms (the \( y's \)) and also use subtraction to isolate the variable on one side, leading to the solution for \( y \). The ability to manipulate equations fluently is fundamental in algebra and enables more complex problem-solving.

Systems of Equations

A system of equations is a set of two or more equations with the same variables that you solve together. The solution to a system is the point or points where the graphs of the equations intersect—that is, the values of the variables that satisfy all equations in the system simultaneously.

Systems can have one solution (where lines intersect at one point), no solution (parallel lines that never meet), or infinitely many solutions (the same line, or overlapping lines). In the case of our problem, we have a system with two linear equations for which the substitution method reveals one unique solution, demonstrating that the lines intersect at exactly one point.