After simplifying, your equation should look like $A x = B$, where $A$ and $B$ each involve $c$. Think about what values of $A$ and $B$ make such an equation have no solution.

A linear equation in $x$ has no solution when the $x$ term cancels out (coefficient becomes 0) but the remaining statement is false (a nonzero constant equals 0). For which value(s) of $c$ does the coefficient of $x$ become 0, and what happens to the constant term then?

For each answer choice, replace $c$ with that value and enter the left and right sides as two separate functions in Desmos, for example for a given $c$:

• $y_1 = (c+2)x + 5$
• $y_2 = 4x + 5c$ (With $c$ replaced by the specific number you are testing.)

For each tested value of $c$, look at the graphs of $y_1$ and $y_2$:

• If the lines intersect at exactly one point, the equation has one solution.
• If the lines overlap completely (same line), there are infinitely many solutions.
• If the lines are parallel but distinct (same slope, different intercepts), there is no solution. Identify which choice for $c$ makes the two lines parallel and distinct; that is the correct answer.

Start with the given equation:

Subtract $4x$ from both sides and subtract $5$ from both sides to group $x$-terms and constants:

Simplify the left side:

The simplified equation is:

For a linear equation of the form $Ax = B$:

• If $A \neq 0$, there is exactly one solution $x = \dfrac{B}{A}$.
• If $A = 0$ and $B = 0$, there are infinitely many solutions (every $x$ works).
• If $A = 0$ and $B \neq 0$, there is no solution (you get a false statement like $0 = 5$).

Here, $A = c - 2$ and $B = 5c - 5$. We want the no solution case.

For no solution, we need:

• The coefficient of $x$ to be $0$:

• The constant term to not be $0$:

These two conditions together will give the value of $c$ that makes the equation have no solution.

First solve the coefficient condition:

Now check the constant term with this value:

So when $c = 2$, the equation becomes $0x = 5$, which is impossible and has no solution. Thus, the correct answer is $c = 2$ (choice C).