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Transmission lines have resistance. Higher voltage can "push" more current through them, thus delivering more power to the consumer.

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The power supplied (i.e. the rate at which energy is supplied) by an electricity transmission line is determined by the load NOT, as insinuated by the original answer, by the supply.

For a given load, the greater the supply voltage, the LOWER the resulting load current. So Transformers are used at power stations to increase the transmission voltage in order to MINIMISE the current (not to 'push more current through'!).

Electricity transmission is only possible because of high voltages causing a reduction in the load current, because, otherwise, the resulting voltage drop in the line would be far too great to supply distant loads.

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