section 1.6
solve each system of equations.
1. $ x + 3y = 17 $
$ y = 7 - x $
2. $ 4x - y = -2 $
$ -2x - 3y = 36 $
3. $ 5x - 4y = 25 $
$ -15x + 8y = 70 $
4. $ 6x + 5y = 20 $
$ 4x - 6y = 32 $
5. a landscaping company placed two orders with a local nursery. the first order was for 13 bushes and 4 trees and totaled $487. the second order was for 6 bushes and 2 trees and totaled $232. what was the cost of each bush and each tree?

Question

section 1.6
solve each system of equations.
1. $ x + 3y = 17 $
   $ y = 7 - x $
2. $ 4x - y = -2 $
   $ -2x - 3y = 36 $
3. $ 5x - 4y = 25 $
   $ -15x + 8y = 70 $
4. $ 6x + 5y = 20 $
   $ 4x - 6y = 32 $
5. a landscaping company placed two orders with a local nursery. the first order was for 13 bushes and 4 trees and totaled $487. the second order was for 6 bushes and 2 trees and totaled $232. what was the cost of each bush and each tree?

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Substitute $ y = 7 - x $ into $ x + 3y = 17 $
Substitute $ y $ in the first equation: $ x + 3(7 - x) = 17 $
## Step2: Simplify and solve for $ x $
Expand: $ x + 21 - 3x = 17 $
Combine like terms: $ -2x + 21 = 17 $
Subtract 21: $ -2x = -4 $
Divide by -2: $ x = 2 $
## Step3: Solve for $ y $ using $ y = 7 - x $
Substitute $ x = 2 $: $ y = 7 - 2 = 5 $
# Answer:
$ x = 2 $, $ y = 5 $

### Problem 2:
# Explanation:
## Step1: Solve $ 4x - y = -2 $ for $ y $
Rearrange: $ y = 4x + 2 $
## Step2: Substitute $ y = 4x + 2 $ into $ -2x - 3y = 36 $
Substitute: $ -2x - 3(4x + 2) = 36 $
## Step3: Simplify and solve for $ x $
Expand: $ -2x - 12x - 6 = 36 $
Combine like terms: $ -14x - 6 = 36 $
Add 6: $ -14x = 42 $
Divide by -14: $ x = -3 $
## Step4: Solve for $ y $ using $ y = 4x + 2 $
Substitute $ x = -3 $: $ y = 4(-3) + 2 = -10 $
# Answer:
$ x = -3 $, $ y = -10 $

### Problem 3:
# Explanation:
## Step1: Multiply the first equation by 2
$ 2(5x - 4y) = 2(25) $ → $ 10x - 8y = 50 $
## Step2: Add to the second equation
$ (10x - 8y) + (-15x + 8y) = 50 + 70 $
Simplify: $ -5x = 120 $
Solve for $ x $: $ x = -24 $
## Step3: Substitute $ x = -24 $ into $ 5x - 4y = 25 $
$ 5(-24) - 4y = 25 $ → $ -120 - 4y = 25 $
Add 120: $ -4y = 145 $
Divide by -4: $ y = -\frac{145}{4} = -36.25 $
# Answer:
$ x = -24 $, $ y = -36.25 $

### Problem 4:
# Explanation:
## Step1: Multiply first equation by 3, second by 5
First: $ 3(6x + 5y) = 3(20) $ → $ 18x + 15y = 60 $
Second: $ 5(4x - 6y) = 5(32) $ → $ 20x - 30y = 160 $
## Step2: Multiply first new equation by 2
$ 2(18x + 15y) = 2(60) $ → $ 36x + 30y = 120 $
## Step3: Add to second new equation
$ (36x + 30y) + (20x - 30y) = 120 + 160 $
Simplify: $ 56x = 280 $
Solve for $ x $: $ x = 5 $
## Step4: Substitute $ x = 5 $ into $ 6x + 5y = 20 $
$ 6(5) + 5y = 20 $ → $ 30 + 5y = 20 $
Subtract 30: $ 5y = -10 $
Solve for $ y $: $ y = -2 $
# Answer:
$ x = 5 $, $ y = -2 $

### Problem 5:
# Explanation:
## Step1: Define variables
Let $ b $ = cost of one bush, $ t $ = cost of one tree.
## Step2: Set up equations
First order: $ 13b + 4t = 487 $
Second order: $ 6b + 2t = 232 $
## Step3: Multiply second equation by 2
$ 2(6b + 2t) = 2(232) $ → $ 12b + 4t = 464 $
## Step4: Subtract from first equation
$ (13b + 4t) - (12b + 4t) = 487 - 464 $
Simplify: $ b = 23 $
## Step5: Substitute $ b = 23 $ into $ 6b + 2t = 232 $
$ 6(23) + 2t = 232 $ → $ 138 + 2t = 232 $
Subtract 138: $ 2t = 94 $
Solve for $ t $: $ t = 47 $
# Answer:
Each bush costs $23, each tree costs $47.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Substitute \( y = 7 - x \) into \( x + 3y = 17 \)

Step2: Simplify and solve for \( x \)

Step3: Solve for \( y \) using \( y = 7 - x \)

Step1: Solve \( 4x - y = -2 \) for \( y \)

Step2: Substitute \( y = 4x + 2 \) into \( -2x - 3y = 36 \)

Step3: Simplify and solve for \( x \)

Step4: Solve for \( y \) using \( y = 4x + 2 \)

Step1: Multiply the first equation by 2

Step2: Add to the second equation

Step3: Substitute \( x = -24 \) into \( 5x - 4y = 25 \)

Answer:

Problem 2: