Skill: Identify systems with infinitely many solutions

Two linear equations have infinitely many solutions when they describe the same line. This happens when all the terms in one equation are proportional to the terms in the other.

Consider:

$$\begin{cases} 2x+3y=6\\ 4x+6y=12 \end{cases}$$

Step 1: Compare the equations

The second equation is the first equation multiplied by $2$:

$$2(2x+3y=6)$$

which gives

$$4x+6y=12.$$

So both equations represent the same line.

Step 2: Confirm by eliminating a variable

Multiply the first equation by $-2$:

$$-4x-6y=-12$$

Now add it to the second equation:

$$\begin{aligned} 4x+6y&=12\\ -4x-6y&=-12\\ \hline 0&=0 \end{aligned}$$

The result is the identity $0=0$. This means the equations do not give two different conditions. They are the same condition.

Step 3: Describe the solution

Every point on the shared line is a solution. Solving the first equation for $y$ gives:

$$2x+3y=6$$ $$3y=-2x+6$$ $$y=-\frac{2}{3}x+2.$$

Therefore, the system has infinitely many solutions: every point on

$$y=-\frac{2}{3}x+2$$

solves both equations. For example, $(0,2)$ and $(3,0)$ are both solutions.

When elimination produces $0=0$, the system has infinitely many solutions. By contrast, intersecting lines give one solution, while distinct parallel lines give no solution.