Alma is buying plants and soil for her garden. The soil cost Php 50 per bag and the plants cost Php 80 each. She wants to buy atleast 5 plants. She cannot spend more than Php 700. Write and graph a system of linear inequalities to model all possible solutions to the situation. Let x represents the number of plants and y represents the number of bags of soil.
Questions:
1.) Give three important informations in the problem:
a.) soil cost Php 50 per bag and plants cost Php 80 each
b.) She wants to buy atleast 5 plants
c.) She cannot spend more than Php 700
2.) Write an inequality that represents the number of plants Alma wants to buy
Let x represents the number of plants
From the statement "She wants to buy atleast 5 plants."
x ≥ 5
3.) Write an inequality that represents how much Alma can spend in total
80x + 50y ≤ 700
4.) What will be our system of linear inequalities in two variables?
80x + 50y ≤ 700
5.) Graph the system here and shade the part where the graph overlap
Please see attached file
Get the x and y intercept to plot the graph
if x = 0
80x + 50y ≤ 700
80(0) + 50y = 700
0 + 50y = 700
50y = 700
y = 700/50
y = 14
(0, 14) ; y intercept
if y = 0
80x + 50y ≤ 700
80x + 50(0) = 700
80x + 0 = 700
80x = 700
x = 700/80
x = 8.75
(8.75, 0) ; x intercept
From the statement "She wants to buy atleast 5 plants."
x ≥ 5
Change to equal sign to get the x intercept
x = 5 ; y = 0
(5, 0) ; x intercept
6.) Give the possible number of plants and number of bags of soil Alma can buy (in coordinate form)
From the graph (the area bounded by the color light red), any ordered pair within is the solution to the inequality.
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(7, 1)
(7, 2)
(8, 1)
7.) Is the point (7, 1) a solution to the problem?
80x + 50y ≤ 700
80(7) + 50(1) ≤ 700
560 + 50 ≤ 700
610 ≤ 700 ; true
YES, the point (7, 1) a solution to the problem
8.) Is the point (4, 2) a solution to the problem?
It depends, if we are concern about the graph formed (which is bounded by red color), with respect to the equation x ≥ 5, (4, 2) is NOT a solution to the problem.
BUT if we are not concerned of the equation x ≥ 5, (4, 2) will be outside the (red area), and by calculation, it will give a true answer based on the equation 80x + 50y ≤ 700
80x + 50y ≤ 700
80(4) + 50(2) ≤ 700
320 + 100 ≤ 700
420 ≤ 700 ; true
9.) Can Alma buy 6 plants and 2 bags of soil without exceeding 700 php?
(6, 2) is bounded by the (red color on the graph)
80x + 50y ≤ 700
80(6) + 50(2) ≤ 700
480 + 100 ≤ 700
580 ≤ 700 ; true
YES, Alma can buy 6 plants and 2 bags of soil without exceeding 700 php
10.) Can Alma buy 9 plants and 3 bags of soil without exceeding 700 php?
(9, 3)
80x + 50y ≤ 700
80(9) + 50(3) ≤ 700
720 + 150 ≤ 700
870 ≤ 700 ; false
NO, Alma cannot buy 9 plants and 3 bags of soil without exceeding 700 php
Answers & Comments
Problem:
Alma is buying plants and soil for her garden. The soil cost Php 50 per bag and the plants cost Php 80 each. She wants to buy atleast 5 plants. She cannot spend more than Php 700. Write and graph a system of linear inequalities to model all possible solutions to the situation. Let x represents the number of plants and y represents the number of bags of soil.
Questions:
1.) Give three important informations in the problem:
a.) soil cost Php 50 per bag and plants cost Php 80 each
b.) She wants to buy atleast 5 plants
c.) She cannot spend more than Php 700
2.) Write an inequality that represents the number of plants Alma wants to buy
Let x represents the number of plants
From the statement "She wants to buy atleast 5 plants."
x ≥ 5
3.) Write an inequality that represents how much Alma can spend in total
80x + 50y ≤ 700
4.) What will be our system of linear inequalities in two variables?
80x + 50y ≤ 700
5.) Graph the system here and shade the part where the graph overlap
Please see attached file
Get the x and y intercept to plot the graph
if x = 0
80x + 50y ≤ 700
80(0) + 50y = 700
0 + 50y = 700
50y = 700
y = 700/50
y = 14
(0, 14) ; y intercept
if y = 0
80x + 50y ≤ 700
80x + 50(0) = 700
80x + 0 = 700
80x = 700
x = 700/80
x = 8.75
(8.75, 0) ; x intercept
From the statement "She wants to buy atleast 5 plants."
x ≥ 5
Change to equal sign to get the x intercept
x = 5 ; y = 0
(5, 0) ; x intercept
6.) Give the possible number of plants and number of bags of soil Alma can buy (in coordinate form)
From the graph (the area bounded by the color light red), any ordered pair within is the solution to the inequality.
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(7, 1)
(7, 2)
(8, 1)
7.) Is the point (7, 1) a solution to the problem?
80x + 50y ≤ 700
80(7) + 50(1) ≤ 700
560 + 50 ≤ 700
610 ≤ 700 ; true
YES, the point (7, 1) a solution to the problem
8.) Is the point (4, 2) a solution to the problem?
It depends, if we are concern about the graph formed (which is bounded by red color), with respect to the equation x ≥ 5, (4, 2) is NOT a solution to the problem.
BUT if we are not concerned of the equation x ≥ 5, (4, 2) will be outside the (red area), and by calculation, it will give a true answer based on the equation 80x + 50y ≤ 700
80x + 50y ≤ 700
80(4) + 50(2) ≤ 700
320 + 100 ≤ 700
420 ≤ 700 ; true
9.) Can Alma buy 6 plants and 2 bags of soil without exceeding 700 php?
(6, 2) is bounded by the (red color on the graph)
80x + 50y ≤ 700
80(6) + 50(2) ≤ 700
480 + 100 ≤ 700
580 ≤ 700 ; true
YES, Alma can buy 6 plants and 2 bags of soil without exceeding 700 php
10.) Can Alma buy 9 plants and 3 bags of soil without exceeding 700 php?
(9, 3)
80x + 50y ≤ 700
80(9) + 50(3) ≤ 700
720 + 150 ≤ 700
870 ≤ 700 ; false
NO, Alma cannot buy 9 plants and 3 bags of soil without exceeding 700 php
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