This is admittedly a difficult problem that Reformed theologians have addressed in several different, though related, ways. Louis Berkhof mentions several approches to reconcile the "reality of the temptations of Jesus" with the idea that he could not sin:

• that in the human nature of Christ, as in that of the first Adam, there was the nuda possibilitas peccandi, the bare abstract possibility of sinning (Kuyper);
• that Jesus' holiness was an ethical holiness, which had to come to high development through, and maintain itself in, temptation (Bavinck);
• and that the things with which Christ was tempted were in themselves perfectly lawful, and appealed to perfectly natural instincts and appetites (Vos).

Unfortunately, Kuyper's work does not appear to be available in English (in either complete or summary form); the reference Berkhof gives is "Dict. Dogm., De Christo, II, pp. 59–108." So in the brief examination of several views that follows, we'll refer to John Frame, Herman Bavinck, and Wayne Grudem (who follows Geerhardus Vos).

Different types of possibility and ability

John Frame addresses the temptation and impeccability of Jesus in the context of a larger discussion on the nature of possibility and ability in his Systematic Theology. It is too simplistic, he says, to insist that any particular thing is either "possible" or "impossible," and mentions several categories of possibility. Something might be logically possible, such as the statement, "John read the entire Bible in 30 seconds," but not physically possible. And though it was physically possible for Jesus's bones to have been broken, since they were human bones, it was metaphysically impossible, because God's decree prevented it in order to fulfill prophecy (John 19:36).

On the question of the temptation of Jesus, then, Frame says:

Could Jesus sin? Perhaps the best short answer is that, yes, he was physically and mentally capable of sinning, but no, he was morally incapable, since he was perfectly holy. Could he struggle with temptation? He could struggle against physical obstacles; why not against mental and spiritual ones as well? As a man, and therefore as a divine-human person, he could struggle mentally with Satan's proposals, growing in his understanding of their nature and consequences, maturing in his ability to relate these to the will of his Father (Luke 2:40, 52). He understood, surely, how evil tempts a man, what pleasures, however fleeting, are to be found in sin. Yet he saw all of these in their true perspective and rejected them.

Development of ethical holiness

Herman Bavinck sees the temptation of Jesus as a means of manifesting his innate holiness. The holiness of his human nature was not like the holiness of his divine nature:

The goodness or holiness of Christ according to his human nature is not a divine and original holiness but one that has been given, infused, and for that reason it must also—in the way of struggle and temptation—reveal, maintain, and confirm itself. Infused goodness does not rule out acquired goodness.

Good fruit, Bavinck says, must come from a good tree, but the goodness of the tree is demonstrated by the goodness of the fruit. "Similarly," he argues:

Christ had to manifest his innate holiness through temptation and struggle; this struggle is not made redundant or vain by virtue of the inability to sin. For although real temptation could not come to Jesus from within but only from without, he nevertheless possessed a human nature, which dreaded suffering and death. [...] And in those temptations he was bound, fighting as he went, to remain faithful; the inability to sin was not a matter of coercion but ethical in nature and therefore had to be manifested in an ethical manner.

Facing temptation without relying on the divine nature

Wayne Grudem, following Geerhardus Vos, argues that Jesus faced and conquered temptation in his human nature alone. Grudem contextualizes the difficulty in our understanding of the hypostatic union, and concludes that thanks to his divine nature, it was indeed impossible for Jesus to sin – but that, as the second Adam, Jesus had to rely on his human nature to defeat temptation, just as Adam did:

Jesus met every temptation to sin, not by his divine power, but on the strength of his human nature alone [...]. The moral strength of his divine nature was there as a sort of "backstop" that would have prevented him from sinning in any case (and therefore we can say that it was not possible for him to sin), but he did not rely on the strength of his divine nature to make it easier for him to face temptations.

Grudem cites the example of the temptation to turn stones into bread – he had the ability, thanks to his divine nature, to do this, but by relying on that ability to conquer the temptation, he would have failed the same test that Adam had failed and could not have earned salvation for us.

As for James 1, Grudem argues that this is similar to other distinctions we must make with respect to the hypostatic union:

His divine nature could not be tempted with evil, but his human nature could be tempted and was clearly tempted. How these two natures united in one person in facing temptations, Scripture does not clearly explain to us.

Summary

After his summary of attempts to deal with this difficulty, Berkhof writes:

But in spite of all this the problem remains, How was it possible that one who in concreto, that is, as He was actually constituted, could not sin nor even have an inclination to sin, nevertheless be subject to real temptation?

So ultimately Reformed thinkers admit that there is a mystery here – the explanations they provide are ultimately inadequate to fully explain the scriptural teaching that Jesus was both tempted and unable to sin.

• Bavinck, Reformed Dogmatics, III, §372
• Berkhof, Systematic Theology, 3.2.1.B.2.e
• Frame, Systematic Theology, 820–822
• Grudem, Systematic Theology, 538–539
• Vos, Biblical Theology, 339–42