Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-y=7 \\ 9 x-3 y=21 \end{array}\right. $$

Short answer

The system has infinitely many solutions.

Step-by-step solution

Write the system of equations

The given system of equations can be written as: 1) \(3x - y = 7\) 2) \(9x - 3y = 21\)

Simplify the second equation

Observe equation 2: \(9x - 3y = 21\). Divide the entire equation by 3 to simplify it: \(\frac{9x}{3} - \frac{3y}{3} = \frac{21}{3}\) This simplifies to \(3x - y = 7\).

Compare the equations

Now we have the simplified system: 1) \(3x - y = 7\) 2) \(3x - y = 7\). Both equations are identical, meaning they represent the same line.

Determine the solution

Since both equations are identical, the system has infinitely many solutions. Every point on the line \(3x - y = 7\) is a solution.

Conclude the solution

The system is consistent and has infinitely many solutions.

Key concepts

linear equations

Linear equations are equations of the first degree, meaning the highest power of the variable is one.
They can be written in the form: \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants.
Usually, these equations graph as straight lines on the Cartesian plane. Solving a system of linear equations involves finding the values of variables that satisfy all equations in the system simultaneously.
Systems can have one unique solution, infinitely many solutions, or no solution at all. Understanding the nature of the solutions helps in determining the relationship between the lines represented by the equations.

infinite solutions

In a system of linear equations, having infinite solutions means that both equations represent the same line.
For instance, in the given system: \[ \left\{ \begin{array}{l} 3x - y = 7 \ 9x - 3y = 21 \end{array}\right. \] After simplifying the second equation, we get: \( 3x - y = 7 \). This shows that both equations are identical.

Since they are the same line, any point on this line is a solution to both equations.
Therefore, there are infinitely many solutions. To verify, you can choose any value for \(x\) and solve for \(y\) using the line equation. For example:
If \( x = 0 \ 3(0) - y = 7 \ -y = 7 \ y = -7 \). So, (0, -7) is a solution.
If \( x = 1 \ 3(1) - y = 7 \ 3 - y = 7 \ -y = 4 \ y = -4 \). So, (1, -4) is also a solution.
This demonstrates the consistent and copious nature of solutions in such cases.

inconsistent system

An inconsistent system of equations is one that has no solution.
This happens when the graphs of the equations do not intersect at any point.
In other words, the lines are parallel and distinct.
For example, consider the system: \[ \left\{ \begin{array}{l} x + y = 2 \ x + y = 5 \end{array}\right. \] Here, the slope of both lines is the same but the y-intercepts are different.
Writing them in the slope-intercept form: \( y = -x + 2 \) and \( y = -x + 5 \)
It shows clearly that both lines have the same slope but different intercepts.
Thus, they will never meet, indicating no common solution.
This makes the system inconsistent. Recognizing an inconsistent system early helps in avoiding unnecessary calculations and focusing on correct solution strategies.