Texas conservatives are hoping to win two major Republican run-off elections on Tuesday, for lieutenant governor and attorney general, underscoring the Tea Party’s enduring influence in the state.

Those two races were left undecided after no single candidate crossed the 50 percent threshold in the March 4 Republican primary, setting the stage for what has been a bruising run-off election between establishment candidates and conservative challengers who have the backing of both the Tea Party movement and its Texas star, U.S. Senator Ted Cruz.

The winner of Tuesday's run-offs will go on to face a Democrat in the November elections.

In the race for Lieutenant Governor, Tea Party favorite State Senator Dan Patrick took 41.5 percent of the vote in the four-candidate March primary, and is favored to win on Tuesday. The three-term incumbent, David Dewhurst, finished with 28 percent.

The run-off race for state attorney general features a similar showdown, after Tea Party-backed State Senator Ken Paxton took 44 percent of the primary vote and State Representative Dan Branch won 34 percent.

The current attorney general, Republican Greg Abbott, already has his party's backing to face Democrat Wendy Davis in the race for Texas governor in November.

Nationally, the movement for smaller government that takes its name from the 18th century U.S. revolt against British colonial rule has seen its strength wane this year. That does not appear to be the case in Texas.

"Republican politics in Texas has become a race to the right," said Republican strategist Bill Miller. "I do think the Republican Party could be eclipsed by the Tea Party here."