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The part of a sentence "agreed to back" is correct and usable in written English.
You can use it in a sentence when two or more people have come to an agreement or understanding, and one has decided to support or approve of the other. For example, "The board of directors all agreed to back the new business venture."
Exact(60)
Delegates also agreed to back indicative ballots in neighbouring schools.
Eventually, the amigos agreed to back a Democrat, but for a price.
Cajoled by their leader, most of Forza Italia's senators agreed to back the government.
VandeHei told Allbritton about his and Harris's idea, which Allbritton agreed to back.
The Greens and the Left Party had agreed to back her.
Scardino has agreed to back plans for an Asian edition of the paper.
The United States and Europe also agreed to back Iran's membership in the World Trade Organization.
And the nation's largest beryllium producer had agreed to back the new restrictions.
According to Kathimerini, the Independent Greeks have agreed to back Syriza's economic plans.
It is understood that privately he had already agreed to back Blatter.
Singapore Airlines and Brierley have agreed to back this plan, the government said.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com