Question 1173100: Mr.Agoncillo has a savings account in two banks. The combined amount of these savings is at least Php 150,000. One bank gives an interest of 4% while the other bank give 6%. In a year, Mr. Agoncillo receive at most Php. 12,000.
Write and graph a system of linear inequalities that shows all the posible solutions.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem and set up the system of linear inequalities.
**Variables**
* Let 'x' represent the amount of savings in the bank with 4% interest.
* Let 'y' represent the amount of savings in the bank with 6% interest.
**Inequalities**
1. **Combined Savings:**
* The combined amount of savings is at least Php 150,000.
* Equation: x + y ≥ 150,000
2. **Interest Earned:**
* The total interest earned is at most Php 12,000.
* Interest from 4% bank: 0.04x
* Interest from 6% bank: 0.06y
* Equation: 0.04x + 0.06y ≤ 12,000
3. **Non-negative Savings:**
* Savings cannot be negative.
* Equations: x ≥ 0 and y ≥ 0
**System of Linear Inequalities**
* x + y ≥ 150,000
* 0.04x + 0.06y ≤ 12,000
* x ≥ 0
* y ≥ 0
**Graphing the Inequalities**
1. **x + y ≥ 150,000**
* To graph this, first graph the line x + y = 150,000.
* Find the intercepts:
* If x = 0, y = 150,000
* If y = 0, x = 150,000
* Draw the line connecting (150000,0) and (0,150000). Since the inequality is "greater than or equal to," use a solid line.
* Shade the region above the line, as that represents x + y ≥ 150,000.
2. **0.04x + 0.06y ≤ 12,000**
* To graph this, first graph the line 0.04x + 0.06y = 12,000.
* To make it easier to graph, we can multiply the entire equation by 100
* 4x+6y=1200000
* Then we can divide the entire equation by 2
* 2x+3y=600000
* Find the intercepts:
* If x = 0, 3y = 600000, y = 200,000
* If y = 0, 2x = 600000, x = 300,000
* Draw the line connecting (300000, 0) and (0, 200000). Since the inequality is "less than or equal to," use a solid line.
* Shade the region below the line, as that represents 0.04x + 0.06y ≤ 12,000.
3. **x ≥ 0 and y ≥ 0**
* These inequalities restrict the solution to the first quadrant (where both x and y are non-negative).
**Finding the Solution Region**
* The solution region is the area where all shaded regions overlap in the first quadrant.
* The corner points of this region are important for finding specific solutions.
**Corner Points**
* Intersection of x + y = 150,000 and x = 0: (0, 150,000)
* Intersection of x + y = 150,000 and 0.04x + 0.06y = 12,000:
* Solve the system of equations.
* From x + y = 150,000, y = 150,000 - x.
* Substitute into 0.04x + 0.06(150,000 - x) = 12,000
* 0.04x + 9,000 - 0.06x = 12,000
* -0.02x = 3,000
* x = -150000. There is an error.
* multiply 0.04x + 0.06y = 12000 by 100. 4x+6y=1200000, then divide by 2. 2x+3y=600000
* multiply x+y=150000 by 2. 2x+2y=300000. Substract that from 2x+3y=600000. y=300000. x=-150000. There is still an error.
* Multiply x+y=150000 by 0.04. 0.04x+0.04y=6000.
* Subtract that from 0.04x+0.06y=12000. 0.02y=6000. y=300000. x=150000-300000. x=-150000. There is still an error.
* 0.04x + 0.06(150000-x)=12000
* 0.04x +9000-0.06x=12000
* -0.02x=3000
* x=-150000
* There is an error with the problem itself. The intersection of those two lines is outside of the first quadrant.
* Intersection of 0.04x + 0.06y = 12,000 and y = 0: (300,000, 0)
* Intersection of x=0 and y=0 (0,0)
The solution region is the area between the lines, and within the first quadrant.
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