One can try to live a good life through good works/deeds and obedience to God's laws
No, not 'try' we must do it.
It is not God's law according to the old law, but Jesus' teachings in the new law.
and still end up being judged by their works and cast into the Lake of Fire for eternity.
How do you ever get that God would throw someone in the lake of fire if they did what He says to do?
1 Peter 1:13 Who is going to harm you if you are eager to do good?
Proverbs 12:21
No harm overtakes the righteous, but the wicked have their fill of trouble.
Titus 2:14
who gave himself for us to redeem us from all wickedness and to purify for himself a people that are his very own, eager to do what is good.
Did you read those scriptures?
Did you read what Titus says?
Jesus GAVE HIMSELF for those who are EAGER TO DO WHAT IS GOOD.
Read what John 7:18 says.
That scripture says there is NOTHING FALSE in the one who seeks the glory of God.
In order for one to be "truly saved," they must have heard the true Gospel, placed their faith in Christ as their Savior, been sealed and indwelt by the Holy Spirit and baptized (not by water) into the Body of Christ by the Holy Spirit. We don't save ourselves, God gives His Grace as a free gift to men of faith.
Jesus saves us all on his own, and he chooses those he saves.
Jesus tells us whom he saves. Jesus SAYS he chooses those who obey him.