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How do British legislative elections help select both a parliament and a prime minister?

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A large crowd of Rwandan Patriotic Front supporters holds up red, white, and blue FPR flags and campaign posters at a political rally in Kayonza, Eastern Province. — political party Rwandan Patriotic Front political rally in Kayonza, Rwanda, August 2010.

In the British parliamentary system, legislative elections serve as the mechanism for selecting both the members of the House of Commons and, indirectly, the prime minister.21

Selecting Parliament

Citizens vote in general elections to elect representatives for individual constituencies using a "first-past-the-post" plurality system. These elected representatives form the House of Commons, which is the primary legislative body in the United Kingdom.234

Selecting the Prime Minister

The selection of the prime minister is an indirect result of these elections. The process functions as follows:5

Party Leadership: Political parties contest the election with a clear leader who acts as the de facto candidate for prime minister.2
Majority Support: By constitutional convention, the prime minister must be a member of the House of Commons and must command the "confidence" (majority support) of that house.14
Appointment: Following an election, the monarch formally appoints the leader of the party that secures a majority of seats in the House of Commons as the prime minister.5

Because the government relies on the support of the House of Commons to pass legislation and the budget, the prime minister and their cabinet are effectively the leading committee of Parliament, ensuring that the executive and legislative branches remain closely linked.15