How to Algebraically Find the Intersection of Two Lines

Step-by-Step Guide

1. 1

   Step 1: Write the equation for each line with y on the left side.

   If necessary, rearrange the equation so y is alone on one side of the equal sign.
   
   If the equation uses f(x) or g(x) instead of y, separate this term instead.
   
   Remember, you can cancel out terms by performing the same action to both sides.
   
   If you do not know the equations, find them based on the information you have.
   
   Example:
   Your two lines are y=x+3{\displaystyle y=x+3} and y−12=−2x{\displaystyle y-12=-2x}.
   
   To get y alone in the second equation, add 12 to each side: y=12−2x{\displaystyle y=12-2x}
2. 2

   Step 2: Set the right sides of the equation equal to each other.

   We're looking for a point where the two lines have the same x and y values; this is where the lines cross.
   
   Both equations have just y on the left side, so we know the right sides are equal to each other.
   
   Write a new equation that represents this.
   
   Example:
   We know y=x+3{\displaystyle y=x+3} and y=12−2x{\displaystyle y=12-2x}, therefore x+3=12−2x{\displaystyle x+3=12-2x}. , The new equation only has one variable, x.
   
   Solve this using algebra, by performing the same operation on both sides.
   
   Get the x terms on one side of the equation, then put it in the form x = __. (If this is impossible, skip down to the end of this section.) Example: x+3=12−2x{\displaystyle x+3=12-2x} Add 2x{\displaystyle 2x} to each side: 3x+3=12{\displaystyle 3x+3=12} Subtract 3 from each side: 3x=9{\displaystyle 3x=9} Divide each side by 3: x=3{\displaystyle x=3}. , Choose the equation for either line.
   
   Replace every x in the equation with the answer you found.
   
   Do the arithmetic to solve for y.
   
   Example: x=3{\displaystyle x=3} and y=x+3{\displaystyle y=x+3} y=3+3{\displaystyle y=3+3} y=6{\displaystyle y=6} , It's a good idea to plug your x-value into the other equation and see if you get the same result.
   
   If you get a different solution for y, go back and check your work for mistakes.
   
   Example: x=3{\displaystyle x=3} and y=12−2x{\displaystyle y=12-2x} y=12−2(3){\displaystyle y=12-2(3)} y=12−6{\displaystyle y=12-6} y=6{\displaystyle y=6} This is the same answer as before.
   
   We did not make any mistakes. , You've now solved for the x-value and y-value of the point where the two lines intersect.
   
   Write down the point as a coordinate pair, with the x-value as the first number.
   
   Example: x=3{\displaystyle x=3} and y=6{\displaystyle y=6} The two lines intersect at (3,6). , Some equations make it impossible to solve for x.
   
   This doesn't always mean you made a mistake.
   
   There are two ways a pair of lines can lead to a special solution:
   If the two lines are parallel, they do not intersect.
   
   The x terms will cancel out, and your equation will simplify to a false statement (such as 0=1{\displaystyle 0=1}).
   
   Write "the lines do not intersect" or no real solution" as your answer.
   
   If the two equations describe the same line, they "intersect" everywhere.
   
   The x terms will cancel out and your equation will simplify to a true statement (such as 3=3{\displaystyle 3=3}).
   
   Write "the two lines are the same" as your answer.
3. 3

   Step 3: Solve for x.

4. 4

   Step 4: Use this x-value to solve for y.

5. 5

   Step 5: Check your work.

6. 6

   Step 6: Write down the x and y coordinates of the intersection.

7. 7

   Step 7: Deal with unusual results.

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