Question

Explain how you can tell if the system of linear equations has a solution. Then solve the system. $$ \begin{aligned}&x-y=2\\\&4 x-4 y=8\end{aligned} $$

Short answer

The system of linear equations has infinitely many solutions in the form \( ( y+2 , y) \), where \( y \) is any real number.

Step-by-step solution

Examine the Equations

Compare the two equations. It can be observed that the second equation is nothing but the first equation multiplied by four. Thus, \(4x - 4y = 8\) is the same line as \(x - y = 2\).

Solve One of The Equations

Choose one equation to solve for a specific variable. Here, choose the simpler equation \(x - y = 2\) to solve for 'x'. By adding 'y' to both sides of the equation, we get: \(x = y + 2\). This means for every value of 'y', 'x' would be 'y' plus two.

Interpret The Result

Since both equations represent the same line, this means the system has infinitely many solutions. Substituting any real number for 'y' will yield a valid 'x', thus the system has an infinite number of solutions in the form (y+2 , y).

Key concepts

Understanding Linear Equations

When we talk about linear equations, we're referring to mathematical statements that show the relationship between two variables generally represented as 'x' and 'y'. They're linear because, when plotted on a graph, they form a straight line. A linear equation in two variables can be represented in the form of \(ax + by = c\), where 'a', 'b', and 'c' are constants.

Let's consider our exercise problem where we have \(x - y = 2\). This is a simple linear equation with 'x' and 'y' as our variables. In a linear system, we could have one solution, no solutions, or infinitely many solutions. Understanding the relationship between the different equations in a system is crucial to identifying which one applies.

Infinite Solutions in Linear Systems

Infinite solutions occur in a system of linear equations when all the solutions for one equation are also solutions for the other equations in the system. This usually happens when the equations are multiples of each other—meaning, they have the same slope and y-intercept, thus representing the same line on a graph.

In the given step-by-step solution, the second equation \(4x - 4y = 8\) is a multiple of the first equation \(x - y = 2\). If we divide every term of the second equation by 4, we'll end up with the first equation. This tells us that they are actually the same line and therefore, have an infinite number of overlapping solutions. Any point on this line satisfies both equations simultaneously.

Utilizing Algebraic Methods to Solve Systems

Algebraic methods involve manipulation of equations in order to solve for unknown variables. Common techniques include substitution, elimination, and graphing. Substitution, for example, involves rearranging one equation to solve for one variable and then substituting that value into the other equation. Elimination involves adding or subtracting equations from each other to cancel out one variable, leaving a single equation to solve.

In our exercise, the substitution method helps us understand the relationship between 'x' and 'y' in a clear manner. By solving one equation for 'x' and substituting that expression into any other equations, we can find a general solution for 'x' in terms of 'y'. However, because we determined that there are infinite solutions, we won't be finding a single, unique pair of 'x' and 'y' values. Instead, we describe the solutions as being all the points on a line, which can be represented as the equation of that line, such as \(x = y + 2\).