Look at the coefficients of $x$ and $y$ in both equations. By what factor do you multiply $2x + 3y$ to get $4x + 6y$?

Once you know the factor that turns $2x + 3y$ into $4x + 6y$, apply that same factor to the constant $15$ on the right side to find $k$.

Type 2x + 3y = 15 into Desmos so you can see the line represented by the first equation.

Type 4x + 6y = k. Desmos will create a slider for k, allowing you to adjust the value of $k$ and see how the second line moves.

Drag the slider for $k$ until the second line lies exactly on top of the first line (they become a single line with the same graph). The value of $k$ at that moment is the one that makes the system have infinitely many solutions—read that value from the slider.

For two linear equations in $x$ and $y$ to have infinitely many solutions, they must represent the same line. That means the coefficients of $x$, the coefficients of $y$, and the constant terms must all be in the same ratio (all multiplied by the same number).

Look at the given system:

Compare the $x$-coefficients and the $y$-coefficients:

• From $2x$ to $4x$, we multiply by $2$.
• From $3y$ to $6y$, we also multiply by $2$.

So the entire left side of the second equation is obtained by multiplying the left side of the first equation by $2$. For the two equations to represent the same line, the constant term on the right side must also be multiplied by this same factor of $2$.

Since the constant term in the first equation is $15$, and the second equation’s left side is $2$ times the first, the constant term $k$ must be $2$ times $15$:

So the value of $k$ that makes the system have infinitely many solutions is 30.