Question

1 ( 3x + 2y = 11 )
( 7x - y = 3 )
( (1, 4) )
2 ( 3x - 4y = 18 )
( x + 3y = -7 )
( (-2, -3) ) (handwritten)
3 ( 5x + 2y = -8 )
( 9x - 4y = -22 )
( (-2, 1) ) (handwritten)
4 ( x - 5y = 15 )
( 4x - 3y = 26 )
( (5, -2) ) (handwritten)
5 ( 2x + 5y = 11 )
( -3x + 8y = -1 )
(handwritten start of coordinate)
6 ( 7x - 3y = 2 )
( 5x + 4y = -17 )
7 ( 4x - 5y = -28 )
( -9x - 2y = 10 )
8 ( 2x + 3y = 10 )
( 3x - 10y = 15 )
9 ( -7x + 4y = -6 )
( 2x - 5y = 21 )
10 ( 8x + 3y = -12 )
( 6x + 5y = -20 )
11 ( -4x - 9y = 1 )
( -x + 2y = -4 )
12 ( 5x - 12y = -16 )
( -3x + 4y = 0 )
13 an algebra teacher drove by a farmyard full of chickens and pigs. the teacher happened to notice that there were a total of 100 heads and 270 legs. how many chickens were there? how many pigs were there?

Explanation:

Step1: Define Variables

Let \( x \) be the number of chickens and \( y \) be the number of pigs.

Step2: Set Up Equations

• Heads: Each chicken and pig has 1 head, so \( x + y = 100 \).
• Legs: Chickens have 2 legs, pigs have 4 legs, so \( 2x + 4y = 270 \). Simplify to \( x + 2y = 135 \).

Step3: Solve the System

Subtract the first equation from the second: \((x + 2y) - (x + y) = 135 - 100\) → \( y = 35 \).
Substitute \( y = 35 \) into \( x + y = 100 \): \( x = 100 - 35 = 65 \).

Answer:

Chickens: 65, Pigs: 35