My grandfather's farm has pigs and chickens. I counted a total of 70 legs and 26 heads. How many...

Question:

My grandfather's farm has pigs and chickens. I counted a total of {eq}70 {/eq} legs and {eq}26 {/eq} heads.

How many chickens does my grandfather have?

Elimination Method:

In the elimination method, we solve a system of two equations of two variables by adding or subtracting the equations. By doing so, we get a linear equation in one variable, which we can solve easily.

Let us assume the number of pigs and chickens to be {eq}p {/eq} and {eq}c {/eq} respectively.

Since there are {eq}26 {/eq} heads in total,

$$p+c = 26 \,\,\,\,\,\,\rightarrow (1)$$

Since there are {eq}70 {/eq} legs in total, and since a pig has 4 legs and a chicken has 2 legs, we have:

$$4p+2c = 70 \,\,\,\,\,\,\rightarrow (2)$$

Multiply both sides of (1) by {eq}-2 {/eq}:

$$-2(p+c=26) \Rightarrow -2p-2c = -52 \,\,\,\,\,\,\rightarrow (3)$$

$$(4p+2c )+(-2p-2c)= 70+(-52) \\ (4p-2p)+(2c-2c) = 70-52\\ 2p= 18 \\ \text{Dividing both sides by 2}, \\ p=9$$

Substitute this in (1):

$$9+c =26 \\ \text{Subtracting 9 from both sides}, \\ c = 17$$

Therefore the number of chickens with grandfather is {eq}\boxed{\mathbf{17}} {/eq}.