5. solve the system of equations shown below graphically. make sure to label each line that you graph with its equation.
$y = -x + 14$
$y = \frac{2}{3}x - 1$
(9,5)
graph
using your math
6. a rectangle has dimensions given by the variables $w$ and $l$, for width and length. the rectangle has a perimeter of 34 feet and an area of 60 square feet.
(a) write a system of two equations involving $l$ and $w$ based on the perimeter and area information.
$w(2) + l(2) = 34$
rectangle diagram
(b) do the values $w = 5$ ft and $l = 12$ feet solve this system? justify.
the values
reasoning
7. explain why the system comprised of the equations $y = 2x + 7$ and $y = 2x - 3$ will not have any solutions.

Question

Accepted Answer

### Question 5
# Explanation:
## Step1: Recall the graphical solution method
To solve a system of linear equations graphically, we find the point of intersection of the two lines. Each line is represented by its equation.

## Step2: Analyze the given graph
The two lines are $ y = -x + 14 $ and $ y=\frac{2}{3}x - 1 $. From the graph, the point where these two lines intersect is $(9,5)$. We can verify this by substituting $ x = 9 $ into both equations:
- For $ y=-x + 14 $, $ y=-9 + 14=5 $
- For $ y=\frac{2}{3}x-1 $, $ y=\frac{2}{3}(9)-1=6 - 1 = 5 $

# Answer:
The solution to the system of equations $ y=-x + 14 $ and $ y=\frac{2}{3}x-1 $ is $(9,5)$

### Question 6 (a)
# Explanation:
## Step1: Recall the formulas for perimeter and area of a rectangle
The perimeter $ P $ of a rectangle is given by $ P = 2(L + W) $, where $ L $ is the length and $ W $ is the width. The area $ A $ of a rectangle is given by $ A=L\times W $

## Step2: Substitute the given values
We know that the perimeter $ P = 34 $ feet and the area $ A = 60 $ square feet.
- For the perimeter: $ 2(L + W)=34 $ (dividing both sides by 2, we get $ L + W=17 $)
- For the area: $ L\times W = 60 $

# Answer:
The system of equations is $\begin{cases}2(L + W)=34\LW = 60\end{cases}$ (or $\begin{cases}L + W=17\LW=60\end{cases}$)

### Question 6 (b)
# Explanation:
## Step1: Check the perimeter equation
Substitute $ L = 12 $ and $ W = 5 $ into the perimeter equation $ 2(L + W) $
$ 2(12 + 5)=2\times17 = 34 $, which matches the given perimeter.

## Step2: Check the area equation
Substitute $ L = 12 $ and $ W = 5 $ into the area equation $ LW $
$ 12\times5 = 60 $, which matches the given area.

# Answer:
Yes, the values $ W = 5 $ ft and $ L = 12 $ ft solve the system. Because when we substitute these values into the perimeter equation $ 2(L + W) $, we get $ 2(12 + 5)=34 $ (which is the given perimeter), and when we substitute into the area equation $ LW $, we get $ 12\times5 = 60 $ (which is the given area).

### Question 7
# Explanation:
## Step1: Recall the slope - intercept form of a line
The slope - intercept form of a line is $ y=mx + b $, where $ m $ is the slope and $ b $ is the y - intercept.

## Step2: Analyze the slopes and y - intercepts of the given lines
For the line $ y = 2x+7 $, the slope $ m_1 = 2 $ and the y - intercept $ b_1 = 7 $
For the line $ y = 2x-3 $, the slope $ m_2 = 2 $ and the y - intercept $ b_2=-3 $

Since the slopes of the two lines are equal ($ m_1=m_2 = 2 $) and the y - intercepts are different ($ b_1
eq b_2 $), the two lines are parallel. Parallel lines never intersect, and the solution of a system of linear equations is the point of intersection of the two lines. So, a system of parallel lines (with different y - intercepts) has no solution.

# Answer:
The system of equations $ y = 2x + 7 $ and $ y=2x - 3 $ has no solution because the two lines have the same slope ($ m = 2 $) and different y - intercepts ($ 7$ and $-3$). Lines with the same slope and different y - intercepts are parallel, and parallel lines do not intersect. Since the solution of a system of linear equations is the point of intersection of the two lines, this system has no solution.

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QUESTION IMAGE

Question

Explanation:

Question 5

Step1: Recall the graphical solution method

Step2: Analyze the given graph

Step1: Recall the formulas for perimeter and area of a rectangle

Step2: Substitute the given values

Step1: Check the perimeter equation

Step2: Check the area equation

Answer:

Question 6 (a)