Question

solve each system by substitution.

1. $-6x + 5y = 20$

$y = -2$

1. shanice bought 9 pairs of pants for a total of $470. dress pants cost $70 and jeans cost $30. how many of each type of pants did she buy?
2. there are 5 animals in the field. some are goats and some are geese. there are 14 legs in all. how many of each animal are in the field?
3. rob wants to buy a parrot for $171.24. he gives the cashier $180. what is his change?

Explanation:

Problem 34

Step1: Substitute \( y = -2 \) into the equation

We have the equation \( -6x + 5y = 20 \) and \( y=-2 \). Substitute \( y = -2 \) into the first equation:
\( -6x + 5\times(-2)=20 \)

Step2: Simplify and solve for \( x \)

Simplify the left - hand side: \( -6x-10 = 20 \)
Add 10 to both sides: \( -6x=20 + 10=30 \)
Divide both sides by - 6: \( x=\frac{30}{-6}=-5 \)

Step1: Define variables

Let \( x \) be the number of dress pants and \( y \) be the number of jeans. We know that \( x + y=9 \) (total number of pants) and \( 70x + 30y=470 \) (total cost). From \( x + y=9 \), we can express \( y = 9 - x \)

Step2: Substitute \( y = 9 - x \) into the cost equation

Substitute \( y=9 - x \) into \( 70x + 30y = 470 \):
\( 70x+30(9 - x)=470 \)
Expand the equation: \( 70x + 270-30x=470 \)
Combine like terms: \( 40x+270 = 470 \)
Subtract 270 from both sides: \( 40x=470 - 270 = 200 \)
Divide both sides by 40: \( x=\frac{200}{40}=5 \)

Step3: Find the value of \( y \)

Since \( y = 9 - x \) and \( x = 5 \), then \( y=9 - 5 = 4 \)

Step1: Define variables

Let \( x \) be the number of goats and \( y \) be the number of geese. We know that \( x + y=5 \) (total number of animals) and \( 4x + 2y=14 \) (total number of legs, since goats have 4 legs and geese have 2 legs). From \( x + y=5 \), we can express \( y = 5 - x \)

Step2: Substitute \( y = 5 - x \) into the leg - equation

Substitute \( y = 5 - x \) into \( 4x+2y = 14 \):
\( 4x+2(5 - x)=14 \)
Expand the equation: \( 4x + 10-2x=14 \)
Combine like terms: \( 2x+10 = 14 \)
Subtract 10 from both sides: \( 2x=14 - 10 = 4 \)
Divide both sides by 2: \( x = 2 \)

Step3: Find the value of \( y \)

Since \( y=5 - x \) and \( x = 2 \), then \( y=5 - 2=3 \)

Answer:

The solution of the system is \( x = - 5,y=-2 \)

Problem 35