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I know that the common type of linking sound is [consonant + vowel]. However, sometimes I hear people link the sounds /s/,/z/,/m/,/n/ with the word "the" /ðə/, /ði:/ by eliminating consonant /ð/, for example: in the /ɪn.ə/, from the /frəm.ə/.

My question is, can we apply this rule when we want to link a consonant (/s/,/z/,/m/,/n/) with any word behind beginning with /ð/ by eliminating /ð/? For example, That is their house /...........ɪz.er............../.

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Khánh Hoàngfor example: in the /ɪn.ə/, from the /frəm.ə/.

If you heard this, then it is poor quality or careless speech, not properly understandable as "in the" and "from the". It is true that the "th" sound can lose some of its full value in fast speech, but if it disappeared altogether we would end up with "in a" and "from a".

Khánh HoàngFor example, That is their house /...........ɪz.er............../.

No, don't say that.