Question
Cherries cost $2/lb. Grapes cost $0.50/lb. You can spend no more than $2 on fruit, and you need at least 2 lb in all.
Create a graph showing the amount of each fruit you can buy.
Give the coordinates of of linear inequalities in two variables.
2x+ y ≤ 4
y≥ -2x-1
Give the coordinates of the system of linear inequalities in two variables.
x+y ≥6
2x-y≤5
Give the coordinates of the solution of the system of linear inequalities.
y < -2x - 3
y > x + 3
Answers
GPT-4o mini
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:**
1. Budget constraint: \( 2x + 0.5y \leq 2 \)
2. Weight constraint: \( x + y \geq 2 \)
3. Non-negativity constraints: \( x \geq 0, y \geq 0 \)
**Rearranging the inequality:**
1. From \( 2x + 0.5y \leq 2 \): Rearranging gives \( y \leq 4 - 4x \).
2. From \( x + y \geq 2 \): Rearranging gives \( y \geq 2 - x \).
### 2. Coordinates of the first set of Linear Inequalities
The inequalities provided are:
1. \( 2x + y \leq 4 \)
2. \( y \geq -2x - 1 \)
**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:
1. \( x + y \geq 6 \)
2. \( 2x - y \leq 5 \)
**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:
1. \( y < -2x - 3 \)
2. \( y > x + 3 \)
**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:
1. **For Cherries and Grapes**: Represented by inequalities, calculate points based on constraints.
2. **First set**: Intercepts found: \( (2, 0), (0, 4), (-0.5, 0), (0, -1) \).
3. **Second set**: Intersection \( \left(\frac{11}{3}, \frac{7}{3}\right) \).
4. **Third set**: Intersection \( (-2, 1) \).
You can plot these points and lines on a graph to visualize the feasible regions.
### 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:**
1. Budget constraint: \( 2x + 0.5y \leq 2 \)
2. Weight constraint: \( x + y \geq 2 \)
3. Non-negativity constraints: \( x \geq 0, y \geq 0 \)
**Rearranging the inequality:**
1. From \( 2x + 0.5y \leq 2 \): Rearranging gives \( y \leq 4 - 4x \).
2. From \( x + y \geq 2 \): Rearranging gives \( y \geq 2 - x \).
### 2. Coordinates of the first set of Linear Inequalities
The inequalities provided are:
1. \( 2x + y \leq 4 \)
2. \( y \geq -2x - 1 \)
**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:
1. \( x + y \geq 6 \)
2. \( 2x - y \leq 5 \)
**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:
1. \( y < -2x - 3 \)
2. \( y > x + 3 \)
**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:
1. **For Cherries and Grapes**: Represented by inequalities, calculate points based on constraints.
2. **First set**: Intercepts found: \( (2, 0), (0, 4), (-0.5, 0), (0, -1) \).
3. **Second set**: Intersection \( \left(\frac{11}{3}, \frac{7}{3}\right) \).
4. **Third set**: Intersection \( (-2, 1) \).
You can plot these points and lines on a graph to visualize the feasible regions.