If x=0 represents 2012 and x=5 represents 2017, the number of subscribers y (in millions) to the two services can be modeled by the linear equations in the following system. Solve this system. Express values as decimals rounded to the nearest tenth. Write the solution set with an ordered pair of the form (year, number of subscribers).
Service A: −5.63x+y=23.8
Service B: 0.71x+y=52.2
To solve the system of linear equations, we can use substitution or elimination method.
We'll use the elimination method to solve the system.
Let's multiply the second equation by -5.63 to make the x coefficients equal:
-5.63 * (0.71x + y) = -5.63 * 52.2
-4x - 5.63y = -294.126
Now we'll write the system of equations:
-5.63x + y = 23.8
-4x - 5.63y = -294.126
To eliminate y, let's add the two equations together:
(-5.63x + y) + (-4x - 5.63y) = 23.8 + (-294.126)
-9.63x - 4.63y = -270.326
Now we have a new equation:
-9.63x - 4.63y = -270.326
Let's multiply the first equation by -4.63 to make the y coefficients equal:
-4.63 * (-5.63x +y) = -4.63 * 23.8
23.8x - 4.63y = -110.994
Now we'll write the system of equations:
23.8x - 4.63y = -110.994
-9.63x - 4.63y = -270.326
Now we will eliminate y by adding the two equations together:
(23.8x - 4.63y) + (-9.63x - 4.63y) = -110.994 + (-270.326)
14.17x - 9.26y = -381.32
Now we have a new equation:
14.17x - 9.26y = -381.32
Let's solve this equation for x:
14.17x = 9.26y - 381.32
x = (9.26y - 381.32) / 14.17
We can substitute this expression for x into one of the original equations and solve for y. Let's use the second equation:
0.71x + y = 52.2
0.71((9.26y - 381.32) / 14.17) + y = 52.2
Now let's solve this equation for y:
0.71 * (9.26y - 381.32) + 14.17y = 52.2 * 14.17
6.5796y - 270.6352 + 14.17y = 738.534
Combine like terms:
20.7496y - 270.6352 = 738.534
Now let's isolate y:
20.7496y = 739.8762
y = 739.8762 / 20.7496
y ≈ 35.7
Now we can substitute this value for y back into the expression for x:
x = (9.26(35.7) - 381.32) / 14.17
x ≈ -3.1
Therefore, the solution to the system of equations is approximately (x, y) = (-3.1, 35.7), which can be written as (-3.1, 35.7) as an ordered pair representing the year and number of subscribers.
To solve the system of linear equations, we can use the method of substitution. Let's solve it step-by-step:
Step 1: Solve the first equation for y in terms of x.
-5.63x + y = 23.8
y = 5.63x + 23.8
Step 2: Substitute the value of y from Step 1 into the second equation.
0.71x + (5.63x + 23.8) = 52.2
6.34x + 23.8 = 52.2
Step 3: Subtract 23.8 from both sides to isolate the variable.
6.34x = 52.2 - 23.8
6.34x = 28.4
Step 4: Divide both sides by 6.34 to solve for x.
x = 28.4 / 6.34
x = 4.5 (rounded to the nearest tenth)
Step 5: Substitute the value of x from Step 4 into the first equation to solve for y.
y = 5.63(4.5) + 23.8
y = 25.335 + 23.8
y = 49.135 (rounded to the nearest tenth)
The solution to the system of equations is x = 4.5 and y = 49.1 (rounded to the nearest tenth).
So, the solution set with an ordered pair of the form (year, number of subscribers) is (2016.5, 49.1).
To solve this system of linear equations, we'll use the method of substitution. We'll start by solving one of the equations for one variable and then substituting that expression into the other equation.
Let's solve the first equation for y:
-5.63x + y = 23.8
y = 5.63x + 23.8
Now, substitute the expression for y into the second equation:
0.71x + (5.63x + 23.8) = 52.2
Combine like terms:
6.34x + 23.8 = 52.2
Subtract 23.8 from both sides:
6.34x = 28.4
To solve for x, divide both sides by 6.34:
x = 28.4 / 6.34
x ≈ 4.49
Now, substitute the value of x back into either equation (let's use the first equation):
-5.63x + y = 23.8
-5.63(4.49) + y = 23.8
Multiply -5.63 by 4.49 and add 5.63(4.49) to both sides:
-25.3 + y = 23.8 + 25.3
-25.3 + y = 49.1
Add 25.3 to both sides:
y = 49.1 + 25.3
y = 74.4
Therefore, the solution to the system of equations is x = 4.49 and y = 74.4. So the ordered pair representing the year and the number of subscribers is approximately (4.5, 74.4).