How does the solution change?

Task

In the equations (a)–(d), the solution $x$ to the equation depends on the constant $a$. Assuming $a$ is positive, what is the effect of increasing $a$ on the solution? Does it increase, decrease, or remain unchanged? Give a reason for your answer that can be understood without solving the equation.

1. $x-a = 0$
2. $ax=1$
3. $ax=a$
4. $\displaystyle \frac{x}{a}=1$

IM Commentary

The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.

This task is adapted from Algebra: Form and Function, McCallum et al., Wiley 2010.

Solution

1. Increases. The larger $a$ is, the larger $x$ must be to give 0 when $a$ is subtracted from it.
2. Decreases. The larger $a$ is, the smaller $x$ must be to give 1 when it is multiplied by $a$.
3. Remains unchanged. This equation is obtained from the equation $x=1$ by multiplying both sides by $a$. So the solution is always the same, $x=1$.
4. Increases. The larger $a$ is, the larger $x$ must be to give 1 when it is divided by $a$.