Combine the fact that $y = x$ with the equation $3x - y = 1$. Substitute $y$ with $x$ and solve for $x$.

Once you know the coordinates of the solution point, plug that $x$ and $y$ into the equation $kx + 2y = 6$ and solve for $k$.

In Desmos, enter the equations y = x and y = 3x - 1. Look at their point of intersection; note the $x$- and $y$-coordinates of this point.

Suppose the intersection from step 1 is $(x_0, y_0)$. In Desmos, type the expression k * x_0 + 2 * y_0 and create a slider for k. Adjust the slider until the value of this expression is 6; the corresponding value of k is the solution.

If the solution $(a,b)$ lies on the line $y = x$, then its $x$- and $y$-coordinates are equal.

So we know that

• $b = a$
• or, written with $x$ and $y$, $y = x$ for the solution point.

Use the line $y = x$ together with the second equation $3x - y = 1$.

Substitute $y = x$ into $3x - y = 1$:

So

Thus

Since $y = x$, we also have

So the solution point is

Now use the first equation $kx + 2y = 6$ and substitute $x = \frac{1}{2}$ and $y = \frac{1}{2}$:

Simplify $2\left(\frac{1}{2}\right)$ to $1$:

Subtract $1$ from both sides:

Multiply both sides by $2$:

So the value of $k$ is 10.