Let's break down each of your requests step by step.
1. Graphing the Fruit Purchase Problem
Variables:
- Let \( x \) be the weight of cherries (in lbs).
- Let \( y \) be the weight of grapes (in lbs).
Inequalities:
- Budget constraint: \( 2x + 0.5y \leq 2 \)
- Weight constraint: \( x + y \geq 2 \)
- Non-negativity constraints: \( x \geq 0, y \geq 0 \)
Rearranging the inequality:
- From \( 2x + 0.5y \leq 2 \): Rearranging gives \( y \leq 4 - 4x \).
- From \( x + y \geq 2 \): Rearranging gives \( y \geq 2 - x \).
2. Coordinates of the first set of Linear Inequalities
The inequalities provided are:
- \( 2x + y \leq 4 \)
- \( y \geq -2x - 1 \)
To find the points of intersection:
- Set \( 2x + y = 4 \) and \( y = -2x - 1 \) to find where they intersect.
Substituting \( -2x - 1 \) into the first equation:
\[ 2x + (-2x - 1) = 4 \] \[ -1 = 4 \quad \text{(not possible)} \]
So, they do not intersect, but we can find their intercepts:
-
For \( 2x + y = 4 \):
- \( x \)-intercept: \( (2, 0) \)
- \( y \)-intercept: \( (0, 4) \)
-
For \( y = -2x - 1 \):
- \( x \)-intercept: \( (-0.5, 0) \)
- \( y \)-intercept: \( (0, -1) \)
3. Coordinates of the second system of Linear Inequalities
The inequalities provided are:
- \( x + y \geq 6 \)
- \( 2x - y \leq 5 \)
To find the points of intersection:
- Set \( x + y = 6 \) and \( 2x - y = 5 \) to find the intersection.
Substituting \( y = 6 - x \) into the second equation:
\[ 2x - (6 - x) = 5 \] \[ 3x - 6 = 5 \] \[ 3x = 11 \implies x = \frac{11}{3} \]
Now plug \( x \) back to find \( y \):
\[ y = 6 - \frac{11}{3} = \frac{18}{3} - \frac{11}{3} = \frac{7}{3} \]
Coordinates:
- Intersection point: \( \left(\frac{11}{3}, \frac{7}{3}\right) \)
Other intercepts can be calculated similarly for each inequality.
4. Coordinates of the solution of the third system of Linear Inequalities
The inequalities provided are:
- \( y < -2x - 3 \)
- \( y > x + 3 \)
To find the points of intersection:
- Set \( y = -2x - 3 \) and \( y = x + 3 \).
Setting them equal:
\[ -2x - 3 = x + 3 \] \[ -3 - 3 = 3x \] \[ -6 = 3x \implies x = -2 \]
Now plug \( x \) back to find \( y \):
\[ y = -2(-2) - 3 = 4 - 3 = 1 \quad \text{or} \quad y = -2 + 3 = 1 \]
Coordinates:
- Intersection point: \( (-2, 1) \)
Summary of Coordinates:
- For Cherries and Grapes: Represented by inequalities, calculate points based on constraints.
- First set: Intercepts found: \( (2, 0), (0, 4), (-0.5, 0), (0, -1) \).
- Second set: Intersection \( \left(\frac{11}{3}, \frac{7}{3}\right) \).
- Third set: Intersection \( (-2, 1) \).
You can plot these points and lines on a graph to visualize the feasible regions.
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