Question

Show that the linear equations $3x - y = 2$ and $9x - 3y = 6$ have many solutions.

Hint:

If one linear equation is a multiple of the other, both represent the same line and hence have infinitely many solutions.

Answer 1

Step 1: Write the given equations.
Equation (1): $3x - y = 2$
Equation (2): $9x - 3y = 6$
Step 2: Simplify equation (2).
Divide both sides of equation (2) by 3: \[ \frac{9x - 3y}{3} = \frac{6}{3} \Rightarrow 3x - y = 2 \] Step 3: Observe.
We see that equation (2) becomes identical to equation (1).
Step 4: Conclusion.
Hence, both equations represent the same line. When two linear equations represent the same line, they have infinitely many solutions. Therefore, the given system of equations has infinitely many (or many) solutions.