Problem 60 - Elementary Algebra

Question

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} y=x+1 \\ -x+y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

Infinitely many solutions; consistent and dependent.

Step 1: Write Down the System of Equations

The given system of equations is: \[ y = x + 1 \] \[ -x + y = 1 \]

Step 2: Compare the Forms

Notice that the first equation is already in the form \( y = mx + b \). Substitute the expression for \( y \) from the first equation into the second equation.

Step 3: Substitute and Simplify

Substitute the first equation into the second equation and simplify: \[ -x + (x + 1) = 1 \]\[ -x + x + 1 = 1 \]\[ 1 = 1 \]

Step 4: Determine the Nature of the Solution

Since the result \(1 = 1\) is a true statement and there is no contradiction, it implies that the two equations represent the same line. Therefore, there are infinitely many solutions where the two lines overlap.

Step 5: Classify the System

The system of equations is consistent and dependent because there are infinitely many solutions as the lines coincide.

Key Concepts

consistent system

A consistent system is one where at least one set of values satisfies all equations in the system. A simple way to understand a consistent system is to think of it as having solutions that 'work' for all equations involved. For instance, if you're working with two equations that intersect at any point or overlap completely, you can find values for the variables that satisfy both equations simultaneously.
In summary, if you can find even one solution that works for all the equations, you have a consistent system.

dependent system

In a dependent system, the equations are not just related; they are essentially the same equation expressed differently. This means one equation can be derived from the other through algebraic manipulation.
For example, if you have the equations:
\y = x + 1 and
\-x + y = 1,
you can see that substituting \( y = x + 1 \) into \-x + y = 1 results in a true statement. This reveals that both equations represent the same line. Thus, a dependent system does not just intersect at a point; they coincide along their entire lengths.
Consequently, a dependent system has infinitely many solutions because every point on one line is also on the other.

infinitely many solutions

When a system has infinitely many solutions, it means there are endless pairs of values that satisfy all equations in the system. Such systems occur when the equations describe the same line or plane.
In the system: \y = x + 1 and \-x + y = 1,
substituting one equation into the other results in an identity like \( 1 = 1 \), indicating that both lines overlap entirely.
Therefore, every point on one line is also a point on the other. Understanding this helps in visualizing that the system doesn't just have one intersection point or a finite number but an infinite number of points satisfying both equations.

substitution method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This transforms a system of equations into a single equation with one variable, making it simpler to solve.
Let's look at the given system:
\y = x + 1
\-x + y = 1.
Step 1: Solve the first equation for \( y \).
It is already in the form: \y = x + 1.
Step 2: Substitute this expression into the second equation:
\-x + (x + 1) = 1.
Step 3: Simplify the equation:
\ -x + x + 1 = 1
\ 1 = 1 - a true statement.
Since the resulting equation is true for all values of \( x \), it indicates that any \( x \) and the corresponding \( y = x + 1 \) will satisfy both equations. Thus, illustrating the system has infinitely many solutions.