After expanding, identify the coefficient of $x$ and the constant term on each side. Think of each side as a line in $x$ with a slope and a vertical intercept.

For an equation of the form $ax+b = cx+d$ to have no solution, what has to be true about $a$ and $c$ (the coefficients of $x$), and what has to be true about $b$ and $d$ (the constants)? Use that to set up an equation involving $k$.

Use your answer from the previous hint to write an equation involving $2k+6$ and $4k+12$. Solve that equation for $k$, then check that the constant terms on the two sides are not equal for that value.

In Desmos, type the expression y = (2x + 6) - (4x + 12). Here, the calculator’s variable x is playing the role of $k$ from the problem. The $x$-values where this graph crosses the $x$-axis are the values of $k$ that make $2k+6$ and $4k+12$ equal.

Tap the point where the graph crosses the $x$-axis (its intercept). Read off the $x$-coordinate of that point; that number is the value of $k$ that makes the coefficients of $x$ on both sides equal.

Using the $k$-value you found, type the corresponding constants into Desmos as separate expressions, for example -5*(that k value) - 8 and 1, and evaluate them. You should see two different numbers, confirming that with this $k$ the equation turns into a false statement and therefore has no solution.

Start with the left side:

Distribute $(k+3)$:

So the whole left side is

Now expand and simplify the constants:

• $2(k+3)x = (2k+6)x$
• $-5(k+3) + 7 = -5k -15 + 7 = -5k - 8$

So the left side becomes

Now simplify the right side:

Distribute $4$:

So the right side becomes

The whole equation is now

Think of each side as a line in $x$:

• Left side: coefficient of $x$ is $2k+6$, constant term is $-5k-8$.
• Right side: coefficient of $x$ is $4k+12$, constant term is $1$.

For an equation of the form

to have no solution, the two lines must be parallel but not the same line. That means:

• The coefficients of $x$ must be equal (same slope).
• The constant terms must be different (different intercepts).

So first, set the coefficients of $x$ equal:

This equation will give the value of $k$ that makes the lines parallel.

Solve the equation from the previous step:

Subtract $2k$ from both sides:

Subtract $12$ from both sides:

Divide by $2$:

Now check the constant terms when $k=-3$:

• Left constant: $-5k - 8 = -5(-3) - 8 = 15 - 8 = 7$.
• Right constant: $1$.

The coefficients of $x$ are equal and the constants are $7$ and $1$, which are different. That makes the equation become a false statement ($7=1$), so it has no solution.

Therefore, the value of $k$ is $-3$.