Find the point of intersection of the lines [tex]$y-x-4=0$[/tex] and [tex]$y=2 x-1$[/tex] using substitution and elimination.

Solve the first equation for y in terms of x : y = x + 4 .
Substitute this expression into the second equation and solve for x : x + 4 = 2 x − 1 ⇒ x = 5 .
Substitute the value of x back into either equation to find y : y = 5 + 4 = 9 .
The point of intersection is ( 5 , 9 ) ​ .

Explanation

Understanding the Problem We have two linear equations: y − x − 4 = 0 and y = 2 x − 1 . We need to find the point of intersection of these two lines using both substitution and elimination methods.

Solving by Substitution Method 1: Substitution


First, solve the first equation for y :
y = x + 4
Next, substitute this expression for y into the second equation: x + 4 = 2 x − 1
Now, solve for x :
x − 2 x = − 1 − 4 − x = − 5 x = 5
Substitute the value of x back into either equation to find y . We'll use the first equation: y = 5 + 4 y = 9

Solving by Elimination Method 2: Elimination

Rewrite the first equation as y − x = 4 and the second equation as y − 2 x = − 1 .
Subtract the second equation from the first equation to eliminate y :
( y − x ) − ( y − 2 x ) = 4 − ( − 1 ) y − x − y + 2 x = 4 + 1 x = 5
Substitute the value of x back into either equation to find y . We'll use the first equation: y − 5 = 4 y = 9

Finding the Point of Intersection Both methods yield the same solution. The point of intersection is ( 5 , 9 ) .

Final Answer Therefore, the point of intersection of the lines y − x − 4 = 0 and y = 2 x − 1 is ( 5 , 9 ) .


Examples
Imagine you are navigating using two different GPS devices, each giving you a linear path to a destination. The point where these paths intersect is where you need to be. Solving systems of equations helps in various fields like economics (finding equilibrium points), engineering (analyzing circuits), and computer graphics (determining intersections of lines and planes).

Answered by GinnyAnswer | 2025-07-04