How may we have Peace with God?

"Therefore, having been justified by faith, we have peace with God through our Lord Jesus Christ" Romans 5:1

'Having been justified by faith', says Paul, 'we have peace with God'. We cannot merit or earn the grace of God. Grace is, by definition, unmerited favor. Likewise, we cannot merit justification by our works. God justifies men freely, and as a gift, without cost. Charles Hodge says this:

"Paul says this peace is the result of justification by faith. He who relies on his works can have no peace. He can neither remove the displeasure of God, nor quiet the apprehension of punishment. Peace is not the result of mere gratuitous forgiveness, but of justification, of a reconciliation founded upon atonement." (Commentary on Romans 5:1).
Justification is based on the work of God's Son. God sent His only Son, Jesus Christ, to live a perfect sinless life, to be a perfect sacrifice, to obtain a perfect redemption so that God reconciles Himself to man. This reconciliation is wrought through the sacrificial death of Jesus Christ on behalf of ungodly sinners; and the imputation of the righteousness of Jesus Christ to them through faith. This great exchange of Jesus Christ doing for sinners what God demands of sinners is called the Substitutionary Atonement.

In Romans 3:24-26, Paul says this:

"being justified as a gift by His grace through the redemption which is in Christ Jesus; whom God displayed publicly as a propitiation in His blood through faith. This was to demonstrate His righteousness, because in the forbearance of God He passed over the sins previously committed; for the demonstration, I say, of His righteousness at the present time, so that He would be just and the justifier of the one who has faith in Jesus." NASB.
The Gospel is the Good News that the righteousness of God has now been revealed. It is the righteousness of God that satisfies His own righteousness in order to receive sinners into His presence. Our redemption is based on a righteousness that is extra nos, that is, apart from us - the righteousness of Christ. Paul goes on to describe the means of Justification - it is by faith and faith alone:
"For we maintain that a man is justified by faith apart from works of the Law." (Romans 3:28). NASB.
And Romans 4:4-5:
"Now to the one who works, his wage is not credited as a favor, but as what is due. But to the one who does not work, but believes in Him who justifies the ungodly, his faith is credited as righteousness" NASB.
All those who take the righteousness of Christ as their own, through faith in His finished work at Calvary, shall be reconciled to God and there shall be no condemnation for them. Through faith in Christ, we appropriate His righteousness. His righteousness is "imputed" (counted) to the believer, and our sin is transferred to Christ in the atonement. That's imputation; that's substitution.