Decisions of the U.S. Supreme Court are formally “opinions” written by the nine members of the court. Many people have opinions on what the law is, and the justices have theirs.

The justices’ opinions count, and when a majority agrees, that decides what the law is.

The decision last week finding the Affordable Care Act constitutional included three different major opinions. Chief Justice John Roberts said that the Commerce Clause of the Constitution, allowing the federal government to regulate domestic trade, does not extend to the failure to buy health insurance. That is not an “activity,” and the clause only covers activities.

Four so-called conservative justices — those appointed by Republican presidents — agreed.

But Roberts said that the taxing power of the federal government allows it to tax people who do not buy health insurance, finding legal the requirement that people must have insurance or pay a tax penalty.

The four so-called liberal justices — those appointed by Democratic presidents — agreed. That was the big decision.

The surprise was that Roberts, himself a Republican appointee, supported the insurance requirement, called the “mandate.”

If you read the 193 pages of the decision, you may get the sense that the justices arrived at their opinions first and then searched for legal reasoning to back them up.

Four wanted the health law to be found unconstitutional, and four wanted it to pass muster. Each side piled up a long list of reasons supporting its position.

Roberts wanted to improve the court’s reputation.

Recent polling shows that many Americans think the U.S. Supreme Court is too political and disapprove of the way it is handling its job. In the June CBS News/New York Times poll, 76 percent thought its decisions were political rather than being based in the law.

As chief justice, Roberts takes responsibility for protecting the court against a loss of public confidence. He made it clear that he was not saying that he thought the health care law was good policy. He was simply saying it was legal.

Earlier last week in a less noted vote, he had also gone with the liberals in rejecting most of Arizona’s immigration law as being unconstitutional.

Even though most Americans do not like the health care mandate, polling showed more positive public attitudes toward the court improved after the decision.

Conservatives seem happy that the court limited the expansion of the Commerce Clause. But there may be less here than meets the eye.

In the extreme Commerce Clause case, in 1942 the court approved regulating an Ohio farmer who produced too much wheat for his own use. That’s a pretty broad interpretation, and it survives.

Twenty-six Republican-controlled states, including Maine, opposed the law and were among the losers in the health care case.

But they got something out of the decision. Roberts and the conservatives ruled the federal government could not cut out existing Medicaid funds for a state refusing to expand that program to cover more people, even though the expansion would be almost fully funded by the federal government.

That puts a burden on the 26 states. Will Maine and the others be able to resist public pressure to expand Medicaid with federal money, just because they don’t like the health law?

Chances seem good that eventually most will not take advantage of the right to reject expanded coverage.

The legal challenge to the health care law was not all that unusual. Social Security faced a similar Supreme Court test. And there may be more legal challenges to parts of the law.

But the main battles will be political. A Republican president and Republicans in control of both houses of Congress could effectively kill the Affordable Care Act. Under the Senate’s strange rules, the federal funding portion of the law can be overturned by a filibuster proof, simple majority.

Republican House Speaker John Boehner has pledged that the Republicans will throw out the entire law. Mitt Romney, the expected GOP presidential candidate, says he wants to keep what people like — no ban on pre-existing conditions, parental coverage for young people, no coverage cap — but to eliminate the part that pays for them — mandatory participation.

Democrats can be expected to defend the law, though President Barack Obama says he is willing to make some changes to it.

Looking at the fate of other game-changing laws, by overcoming early legal and political challenges, they are more likely to survive long term.

People get used to the change and begin to count on it. What the court did last week is to bring the health law closer to something people will learn to live with.