Question 225·Hard·Math

Think about when an equation like $ax + b = cx + d$ is true for every possible $x$. What must be true about $a$ and $c$, and about $b$ and $d$?

After you expand, compare the coefficients of $x$ on both sides to get one equation, and compare the constant terms to get a second equation involving $m$ and $n$.

Once you solve for $m$ and $n$, check each of I, II, and III against those values and see which ones must always hold.

In Desmos, type f(x) = 7x + m and on the next line type g(x) = n(x + 2). Desmos will create sliders for $m$ and $n$.

Remember that an equation has infinitely many solutions when its two sides represent the same line on a graph, so you are looking for $f(x)$ and $g(x)$ to lie exactly on top of each other for all $x$.

Use the sliders for $m$ and $n$ to adjust their values until the graphs of $f(x)$ and $g(x)$ are indistinguishable (they overlap completely).

When the two graphs coincide, look at the slider values of $m$ and $n$—these are the values that make the equation true for all $x$, and you can then check which of I, II, and III match those values.

A linear equation in one variable, like $ax + b = cx + d$, has infinitely many solutions only if it is an identity—that is, both sides are the same expression for every value of $x$.

That happens only when the coefficients of $x$ are equal and the constant terms are equal.

Start with the equation:

Distribute $n$ on the right-hand side:

Now both sides are in the form “(coefficient)$\cdot x$ + constant,” which makes it easy to compare.

For the equation

to be true for all $x$, the $x$-coefficients must be equal and the constant terms must be equal:

• Coefficient of $x$: $7 = n$
• Constant term: $m = 2n$

Substitute $n$ from the first equation into the second to find $m$.

From the coefficient comparison:

• $7 = n$, so statement I ($n = 7$) is true.
• Then $m = 2n = 2 \cdot 7 = 14$, so statement II ($m = 14$) is also true.
• Statement III ($m \neq 14$) directly contradicts $m = 14$, so it must be false.

Therefore, the statements that must be true are I and II only.