**Solving linear equations by elimination methods can be very easy provided we apply the simple rules as explained. To solve an equation by the method of elimination, we should first make a cross-checking on the given equation to see which approach will be preferable. Today, we shall be solving 3 different examples by applying the different approaches. Take a look at the below first example.**

$x\u20134y=15$

$\phantom{\rule{0ex}{0ex}}\mathrm{6x}+4y=6$

**Solve for the value of x and y using the elimination method(Example 1)**

**Looking at the above-given equation, we could easily identify that the variable y, has the same coefficient( which is 4). The signs of the coefficients are opposite to each other (- and +). In this case, this equation can be solved by simply adding both equations together as shown below**

$x\u20134y=15$

$\phantom{\rule{0ex}{0ex}}\frac{+6x+4y=6}{}$

7x=21

**To get the value of x, we will need to divide both sides of the equation by 7. This will imply that x=3.**

**Now, we can substitute x=3 in any of the above equation to get the value of y. **

**choosing x-4y=15, this will be written as 3-4(y)=15 3-4y=15→ -4y=15-3→ -4y=12 **

**→now divide both sides of the equation by -4 to get the value of y. **

**This will implies →y=-3. the solution of this equation is at point (3,-3)**

**Example 2 **

**solve for the value of x and y by elimination method in the equation**

4x+2y=12

6x+2y=8

**This equation is similar to the first equation we just solved. but in this case, we can not directly add both equations since the signs of the y variable in both equations are the same. **

**To solve this kind of equation by the method of elimination, we need to choose any equation and multiply it with a minus sign(-). when this is done, we can then add both equations together as shown below.**

**choosing 4x+2y=12 and multiply by minus sign will imply**

**-(4x+2y=12) this will give us -4x-2y=-12**

**we can now add both equations together. this will give us**

-4x-2y=-12

+ 6x+2y=8

**2x=-4. **

**To get the value of x, we will divide both sides of the equation by 2.**

**it will then implies that x=-2.**

**now substituting x=-2 in any of the above equation. By choosing equation 1 will imply. **

**4x+2y=12→4(-2)+2y=12→-8+2y=12→2y=12+8→2y=20. now divide both sides of the equation by 2 to get the value of y. this will imply that y=10**

**the solution of this equation is at a point where x=-2 and y=10 pt(-2,10)**

Example 3

**Solve for the value of x and y in the below equation by using the method of elimination.**

2x+3y=12

x-y=6

**looking at this equation, neither x or y has the same coefficient. so, this equation **can not** be directly solved by addition.**

**In cases like this, will need to multiply the coefficient of the variable that has an opposite sign on the reverse direction. **

**For example in this equation, the variable y has opposite signs and a coefficient of 3 and 1. will need to multiply the equation by 3 and 1 in the reverse direction**.

**this will implies**

**1(2x+3y=12)**

**3(x-y=6)**

**this will imply**

**2x+3y=12**

**3x-3y=18**

**this equation can now be solved by adding both equations directly**

**by adding both equations will imply**

2x+3y=12

3x-3y=18

5x=30

**by dividing both sides of the equations by 5 to get the value of x, it will imply that **

**x=6**

**now substituted the value of x as 6 in any of the above equations to get the value of y**

**by choosing x-y=6**

**→6-y=6→-y=6-6→-y=o dividing both sides by minus, →y=0**

**so the solution of this equation is at a pion where x=6 and y=0 pt(6,0)**

Check More examples on this web page.