WASHINGTON — The Supreme Court opened its term Monday with new Justice Elena Kagan on the bench — but for only one of the two cases being heard.

Throughout the fall, the pattern will be much the same. Kagan will be deciding about half as many cases as her colleagues, the result of her previous job as the government’s chief lawyer before the high court.

As the U.S. solicitor general, she decided which federal cases would be appealed. Now, however, she is obliged to step aside, or recuse herself, in all of the cases in which she had played a role.

The ninth justice votes last in the court’s private conference, and she is the tie-breaker. But the court will often be without its tie-breaker for the first months of this term, and the justices could find themselves deadlocked and unable to rule in several major cases.

They include the closely watched dispute over Arizona’s crackdown on employers who hire illegal immigrants. A 2007 law threatens to strip such businesses of their license to operate if they hire illegal workers.

The U.S. Chamber of Commerce and the American Civil Liberties Union challenged the state law and argued that only the federal government can enforce immigration rules. The Obama administration agreed in May shortly after President Barack Obama nominated Kagan for the high court.

Though Kagan did not sign the government brief, she nonetheless said she will not participate in the case when it is heard in December. If the court splits 4-4, the Arizona law will stand, since it was upheld last year by the 9th U.S. Circuit Court of Appeals.

On Tuesday, the court will hear three cases, and Kagan will be absent for all for three. They include a privacy case to decide whether the National Aeronautics and Space Administration may require scientists and other contract workers at the Jet Propulsion Lab in Pasadena, Calif., to submit to new background checks and to answer questions about their private lives.

Kagan will help decide this term’s major free-speech disputes involving funeral protests and video games. She could also be a tie-breaker in a death penalty case from Texas where a Death Row inmate is seeking DNA testing of the crime scene evidence. She will also participate in a church-state dispute from Arizona to decide whether the state can give $500 tax credits to those who help pay tuition of students in religious schools.

And her absence is not likely to be felt in many of the routine federal cases involving matters such as taxes, contracting or criminal sentencing. Rarely do the justices split evenly on those issues.

On Monday Kagan got a first glimpse of the mundane aspect of her new job. In her first case, the justices were called upon to decide whether a bankrupt Nevada man could take a $471 per month allowance for a car payment, even though he owned a 2004 Toyota Camry and had no car payment.

Kagan asked six questions of the lawyers in her distinctive New York accent, as many as any of her colleagues.