Question 209·Medium·Math

After simplifying, try to rewrite the equation so it looks like $A x = B$, where $A$ and $B$ are expressions in $k$.

For an equation like $ax = b$, under what condition on $a$ and $b$ is it impossible for any $x$ to make the equation true?

Once you have an equation of the form $(7 - k)x = k - 2$, decide what $7 - k$ must be so that there is no value of $x$ that makes the equation true.

Rewrite the equation as a single expression equal to 0: $5(2x - 1) - (k + 3)x - (k - 7) = 0$. For each answer choice, replace $k$ with that value and type the resulting expression into Desmos as, for example, y = 5(2x - 1) - (2 + 3)x - (2 - 7) (using the chosen value in place of 2).

For a given value of $k$, look at the graph of $y =$ that expression. Any $x$-intercept (where the graph crosses the $x$-axis, so $y = 0$) represents a solution to the equation. If the graph is a horizontal line that never touches the $x$-axis, that value of $k$ gives no solution.

Quickly change the value of $k$ in your expression to each of the four answer choices and observe the graph. Three of them will produce a line that crosses the $x$-axis (so the equation has a solution); the remaining value will produce a line that never crosses the $x$-axis—that is the $k$ for which the equation has no solution.

Start by distributing on the left side.

• Distribute the 5: $5(2x - 1) = 10x - 5$
• Distribute the minus sign: $-(k + 3)x = -kx - 3x$

So the equation becomes:

Now combine the $x$ terms:

So we have:

Move the constant term $-5$ to the right side by adding 5 to both sides:

This simplifies to:

Now the equation is in the form $a x = b$, where $a = 7 - k$ and $b = k - 2$.

For an equation of the form $ax = b$:

• If $a \neq 0$, there is exactly one solution: $x = \dfrac{b}{a}$.
• If $a = 0$ and $b = 0$, every $x$ works (infinitely many solutions).
• If $a = 0$ and $b \neq 0$, there is no solution (you get $0x = \text{nonzero}$, which is impossible).

In our equation $(7 - k)x = k - 2$, we need $7 - k = 0$ so that the $x$ term disappears, and then we will check that the right side is not zero.

Set the coefficient of $x$ equal to 0:

Solving gives $k = 7$.

Substitute $k = 7$ into $(7 - k)x = k - 2$:

The equation $0x = 5$ is impossible for any $x$, so the original equation has no solution when $k = 7$. Thus, the correct answer is 7.