The Addition Method
There are two methods we can use to find the intersection of two lines: the substitution method and the addition method. Of the two, the substitution method, under the right circumstances, is the easier to use but the addition method's advantage is that it works equal well in any situation.
The process for using the substitution method goes like this:
- Pick one of the variables. It doesn't matter which but it's usually easier if you pick ones that have smaller numbers in front of them.
- Find the least common multiple of the coefficients of the variable that you picked.
- Multiply each of the equations by a number that makes the coefficient of the variable you chose in step one equal to the number you found in step two.
- Add or subtract the two equations so that the variable you picked in step one gets canceled.
- Solve the equation that you found in step four. That will give you the value of one of the two variables.
- Substitute the value you found in step five into one of the two original equations. Solve that equation for the second of the two variables.
Example 1
Find the intersection of the lines 2y + 3x = 5 and 5y - 2x = 22.
For this question, I'm going to work on canceling the x terms in the two equations. The coefficients of those variables are 3 and -2. The least common multiple of those numbers is 6. Now, I'll multiply the first equation by 2 (so that the coefficient of the x term becomes 6) and the second equation by 3 (so the coefficient of its x term becomes -6, don't worry about the negative sign).
2 · (2y + 3x = 5) ⇒ 4y + 6x = 10
3 · (5y - 2x = 22) ⇒ 15y - 6x = 66
Notice how the x terms have opposite coefficients. That's what we were aiming for. Now when we add the two equations together, the x terms will go away leaving us an equation with only one variable.
4y + 6x = 10 15y - 6x = 66 |
19y = 76 y = 76 / 19 = 4 |
Now, I'll substitute 4 for y in the first equation to get an equation that I can solve for x.
2 · 4 + 3x = 5
8 + 3x = 5
3x = -3
x = -3 / 3 = -1
This makes the intersection of the two lines the point (-1, 4).
Example 2
Find the intersection of the lines 3y + 4x = 7 and 4y + 2x = 6.
For this question, I'm going to start working with the x's. The least common multiple of 4 and 2 is 4 so I don't need to multiply the first equation by anything because it's x term already has 4 as its coefficient. (That's why I picked the x's to work with on this one.) For the second equation, I'll multiply it by 2 so that its x term's coefficient becomes 4.
2 · (4y + 2x = 6) ⇒ 8y + 4x = 12
Now I'll subtract that equation from the first one. Why subtracting? Remember that the goal is to cancel out the x terms. If I subtract the 4x in one of the equations from the 4x in the other, I'll end up with 0x.
3y + 4x = 7 - (8y + 4x = 12) |
⇒ ⇒ |
3y + 4x = 7 -8y - 4x = -12 |
-5y = -5 y = -5 / -5 = 1 |
Now, I'll substitute 1 for y in the first equation to get an equation that I can solve for x.
4 · 1 + 2x = 6
4 + 2x = 6
2x = 2
x = 2 / 2 = 1
This makes the intersection of the two lines the point (1, 1).
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Directions: This solution has 7 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.