Leka,
I think this is your Jacob Solomon family, clincher being his widowed sister Elizabeth Lister and family living next door to him in Heworth in 1871 census. Plus his mothers name spelling changes to Solomon later in life, same as his.
In chronological order :
A marriage at Chester le Street St. Mary and St. Cuthbert :
23 May 1807. Thomas Salmon and Jane Gray.
Some baptisms at Chester le Street St. Mary and St. Cuthbert :
17 July 1808 John Salmon, of Ouston, born 17 June 1808, 1st son of Thomas Salmon (Pitman, Native of Chester le St.) by his wife Jane Gray (Native of Tanfield)
22 Apr 1810 Isabella Salmon, of Urpeth, born 23 Feb 1810, 1st daughter of Thomas Salmon (Pitman, Native of Great Lumley) by his wife Jane Gray (Native of Tanfield)
28 Feb 1813 Elizabeth Salmon, of Urpeth, daughter of Thomas (Pitman) & Jane Salmon
23 July 1815 Thomas Salmon, of Urpeth, son of Thomas (Pitman) & Jane Salmon
15 Mar 1818 Jacob Salmon, of Urpeth, son of Thomas (Pitman) & Jane Salmon
8 Oct 1820 Peter Donnison Salmon, of Ouston, son of Thomas (Pitman) & Jane Salmon
Then a baptism and burial, same day, at Houghton le Spring St. Michael :
Baptized 3 Oct 1822 Jane Salmon, of Collier Row, daughter of Thomas (Pitman) & Jane Salmon
Buried 3 Oct 1822. Peter Donnison Salmon, aged 2y3m, son of Thomas and Jane Salmon of Collier Row.
Then another baptism same place :
9 Oct 1825 Mary Ann Salmon, of Collier Row, daughter of Thomas (Pitman) & Jane Salmon
Then a burial at Houghton le Spring St. Michael :
16 Aug 1827. Thomas Salmon, aged 49, of Collier Row.
Then 1841 census, Jacob with his mother :
1841 census, Colliery Row, Houghton le Spring
Jane Salmon, 50, Ind, Y,
Jacob Salmon, 20, Coal Miner, Y
Jane Salmon 15, Y
1851 census, Colliery Row, Houghton le Spring
Jane Salmon, 64, Wid, Laundress, b. Hill Top, Durham,
Elizabeth Lister, 39, Dau, Mar, Coal Miners Wife, b. Urpeth
Eliza Lister, GDau, 2 months, b. West Rainton
(Elizabeth Salmon married George Lister, 26 Sep 1835 Houghton le Spring. Widowed Elizabeth Lister and family, including a son Solomon Lister, are living next door to your Jacob Solomon in Heworth in 1871 census)
1861 census, Colliery Row, Houghton le Spring
Jane Solomon, 73, Wid, b. Colliery Dykes, Durham,
Ann Olomon, 23, Visitor, Unm, b. West Rainton
And lastly, another burial at Houghton le Spring St. Michael :
19 Aug 1865. Jane Solomon, aged 80, of Collier Row
Alan.