A cell phone provider offers a plan that costs $40 per month plus $0.20 per text message sent or received. A comparable plan costs $70 per month but offers unlimited text messaging. Complete parts a. and b. below.
(Simplify your answer. Type an integer or a decimal.)
a. Determine the number of text messages for which the two plans cost the same amount during one month.
Let x be the number of text messages sent or received in one month.
For the first plan, the cost is $40 + $0.20x.
For the second plan, the cost is $70.
Setting these equal to each other, we have:
$40 + $0.20x = $70
Subtracting $40 from both sides, we get:
$0.20x = $30
Dividing both sides by $0.20, we get:
x = 150
Therefore, the two plans cost the same amount for 150 text messages sent or received in one month.
b. Determine the number of text messages for which the second plan is cheaper than the first plan during one month.
For the second plan, the cost is always $70, regardless of the number of text messages sent or received.
For the first plan, the cost is $40 + $0.20x.
We need to find the value of x where $70 is less than $40 + $0.20x.
Simplifying the inequality, we have:
$70 < $40 + $0.20x
Subtracting $40 from both sides, we get:
$30 < $0.20x
Dividing both sides by $0.20, we get:
150 < x
Therefore, for the second plan to be cheaper than the first plan, the number of text messages must be greater than 150.
Solve the system using substitution. Check your answer.
y = -5x + 40
3y - x + 8 = 0
(Simplify your answer. Type integers or simplified fractions. Type an ordered pair.)
From the first equation, we can solve for y in terms of x:
y = -5x + 40
Substituting this expression for y in the second equation:
3(-5x + 40) - x + 8 = 0
Simplifying, we get:
-15x + 120 - x + 8 = 0
-16x + 128 = 0
-16x = -128
x = 8
Substituting this value of x into the first equation:
y = -5(8) + 40
y = -40 + 40
y = 0
Therefore, the solution to the system of equations is (8, 0). To check our answer, we can substitute these values into the second equation:
3(0) - 8 + 8 = 0
0 - 8 + 8 = 0
0 = 0
Since both sides of the equation are equal, the solution (8, 0) checks out.
Solve by the substitution method.
9x + 9y = -18
-7x + y = 30 (Type an ordered pair.)
From the second equation, we can solve for y in terms of x:
y = 7x + 30
Substituting this expression for y in the first equation:
9x + 9(7x + 30) = -18
9x + 63x + 270 = -18
72x + 270 = -18
72x = -18 - 270
72x = -288
x = -288/72
x = -4
Substituting this value of x into the second equation:
-7(-4) + y = 30
28 + y = 30
y = 30 - 28
y = 2
Therefore, the solution to the system of equations is (-4, 2). To check our answer, we can substitute these values into both equations:
9(-4) + 9(2) = -18
-36 + 18 = -18
-18 = -18
-7(-4) + 2 = 30
28 + 2 = 30
30 = 30
Since both sides of both equations are equal, the solution (-4, 2) checks out.