Both Democratic front-runner Hillary Clinton and Republican Donald Trump expressed love for the Empire State after they won their respective New York primary elections April 19. (Sarah Parnass/The Washington Post)

Emboldened by dominant victories in New York, Hillary Clinton and Donald Trump plunged swiftly Wednesday into the next batch of primaries in five states along the Northeast Corridor, where they hope to bury or break their challengers for the Democratic and Republican presidential nominations.

Pennsylvania, Maryland, Connecticut, Delaware and Rhode Island will vote Tuesday in what many are dubbing the “Acela primary,” putting Clinton and Trump on terrain well tailored to their campaigns.

For Clinton, it’s a chance to effectively end the long-shot hopes of Sen. Bernie Sanders (Vt.) in the Democratic race.

For Trump, the contests are an opportunity to pad his delegate lead over Sen. Ted Cruz (Tex.) and send him tumbling into the final six weeks of the campaign.

That crucial period will determine whether the mogul will clinch the GOP nomination outright or if the race will head to a contested convention. In a new internal memo, the Trump campaign projected that he will accumulate 1,400 delegates to win the party’s nod on the first ballot.

Donald Trump said that the delegate selection process in the upcoming Pennsylvania Republican primary is unfair at a campaign event in Indianapolis, Ind. on April 20. (Reuters)

Trump faces a key test here in Pennsylvania, Tuesday’s biggest prize. Cruz is making an aggressive push to influence who will become delegates from the Keystone State, which will send most of its slate to the Republican National Convention officially unbound to any candidate. Even a statewide win by Trump could be undercut if he does not derail Cruz’s plan.

After defeating Sanders handily in New York on Tuesday, Clinton planned to talk about gun violence at a predominantly African American church here in Philadelphia on Wednesday afternoon and hold an organizing event in the evening. Clinton is looking to maintain her strong support among black voters next Tuesday. Large urban areas with sizable black populations will be voting.

Trump, who trounced Cruz and Ohio Gov. John Kasich in his home state of New York, held an evening rally in Berlin, Md., near the border with Delaware. Trump has performed very well in Northeast primaries.

Anticipating defeat in New York, Cruz, Kasich and Sanders jumped ahead to campaign in Pennsylvania on Tuesday. The senator from Texas even gave a prime-time speech in Philadelphia, looking past the New York results. Cruz campaigned in Pennsylvania Wednesday, and Kasich planned to do the same.

In the Democratic primary, Clinton is favored to claim Tuesday’s top two trophies: Pennsylvania and Maryland. She has personal ties to the former and frequently talks about her grandfather, who worked in a lace mill in Scranton, Pa.

Clinton talked up her links to Pennsylvania on Wednesday, telling a crowd of about 2,000 in Philadelphia that she was baptized in her father’s home town of Scranton and spent summers there.

“My dad went to Penn State, my brother went to Penn State,” and her son-in-law is from Philadelphia, she said to cheers.

Hillary Clinton won the April 19 New York Democratic primary. Here's how. (Peter Stevenson/The Washington Post)

The applause grew louder when she noted that the Democratic convention will be held in Philadelphia in July and that she hopes to be the nominee.

Clinton was briefly heckled by Black Lives Matter activists, some of whom held up signs as she spoke that read “Hillary Is Killing.”

That was a reference to the omnibus crime bill passed under her husband’s presidency in 1994. The activists claim that the bill has devastated African American communities.

She did not engage the demonstrators, who chanted “don’t vote for Hillary” as they were escorted from the event hall.

Earlier Wednesday, Clinton held a discussion with former attorney general Eric H. Holder Jr. and African American mothers who lost children to shootings. Ahead of the session at a downtown Philadelphia church, protesters shouted outside, demanding that Clinton pardon “victims” of the 1994 crime bill.

Together with New York, the states voting in next week’s Acela primary — named after the high-speed Amtrak train that travels through them — should effectively put the nomination out of reach for Sanders, said former Pennsylvania governor Edward Rendell, a Clinton backer.

“The math will be too hard,” Rendell said. “My guess is she will be up by 300 delegates, and after the Acela primary there are only six, seven, eight opportunities left, and the only really big one is California. That’s not enough.”

But even lopsided victories for Clinton in Pennsylvania, with 189 delegates at stake, and Maryland, with 95, would probably not end Sanders’s candidacy. His aides have continued to argue that there is a path to catching Clinton in the number of delegates awarded in primaries and caucuses — a scenario that would require a series of resounding wins in May and a huge victory in California in June.

Adding to Sanders’s challenge next week are the rules governing the contests. Four of the five states — Pennsylvania, Maryland, Connecticut and Rhode Island — are closed to independents, as New York was. Sanders has done better in states where unaffiliated voters can support him in the nominating contest.

Sanders has said he thinks he will win the Pennsylvania primary. His spokesman, Michael Briggs, said Sanders should also perform well in Connecticut, citing wins in other New England states. He will take a day off the campaign trail on Wednesday before returning in Pennsylvania on Thursday.

On the Republican side, 172 delegates are at stake in the five states voting Tuesday, including 71 from Pennsylvania. Recent polls show Trump leading Cruz in the state.

“I think Mr. Trump’s going to do very well in the state of Pennsylvania,” campaign manager Corey Lewandowski predicted.

In each of the state’s 18 congressional districts, Republican voters will select not just a presidential candidate, but also a slate of three delegates who will go to the Cleveland convention unbound. The statewide Republican winner will get a separate prize of 17 delegates pledged to him on the first ballot at the convention.

Cruz’s campaign plans to persuade voters to elect his preferred delegates in each district.

“We’re going to voters saying: When you go into the polls on Election Day, vote four times for Ted Cruz. Vote for Ted Cruz and vote for each of his three congressional district delegates,” said Lowman Henry, Cruz’s Pennsylvania state chairman.

Henry said the Cruz campaign is “going to make that plain in a variety of ways through really 18 different congressional district campaigns we’re going to be running over the next week.” He declined to lay out the specific strategy for doing so. But there is no guarantee it will work. And if it doesn’t, Pennsylvania could quickly become Trump territory. A recent Pittsburgh ­Tribune-Review survey of the people running for delegate in their district showed many would support the statewide or districtwide winner on the first ballot in Cleveland.

Trump convention manager Paul Manafort said Pennsylvania will be different from previous states where Cruz sailed through the delegate-selection process.

“We were involved in the filing, and we’re going to run a very competitive race,” he said.

Cruz strategists think they can win enough support in the remaining contests to keep Trump from getting to 1,237 delegates, the number needed to clinch the nomination. Trump’s team, meanwhile, has said it is hopeful that he will eclipse that threshold by the time primary season ends in June. After New York, Trump led Cruz 845 delegates to 559, according to the Associated Press’s latest tally Wednesday morning.

But Kasich is a wild card for Trump and Cruz. The Ohio governor’s centrist pitch could win him some support Tuesday’s contests, where many Republicans are moderates. Kasich planned to campaign in at least three of the five states between Wednesday and Saturday. In a memo released Tuesday night, chief strategist John Weaver argued that Kasich was “best positioned” against Trump on Tuesday.

The extended race is straining Cruz’s financial capabilities. Newly released Federal Election Commission filings showed Wednesday that his campaign spent 94 percent of what it raised in March, a high burn rate that comes at a time when many Republican donors have not warmed to Cruz’s campaign.

Both Trump and Cruz have started looking ahead to the May 3 battleground of Indiana, where Cruz’s campaign feels it could end a potential two-week losing streak in the Northeast. Trump’s first scheduled campaign stop Wednesday was a rally in Indianapolis. Cruz was expected to campaign there on Thursday.

Along with Pennsylvania, Cruz’s strategists see Maryland as another state where he could make some headway, but no one is predicting big victories. If anything, they are bracing for another impressive showing by Trump.

“I think once we get by the 26th and we start heading west again, you’re going to particularly there see Senator Cruz start to rack up delegates,” Henry said.

Gearan reported from Washington. John Wagner in State College, Pa., Jenna Johnson in New York and Philip Rucker in Hollywood, Fla., contributed to this report.