Money is invested at two rates of interest. One rate is 8% and the other is 2%. If there is $1000 more invested at 8% than at 2%. Find the amount invested at each rate if the annual interest from both investments is $310. Let x amount invested at 8% and y = amount invested at 2%. Then the system that models the problem is [x=y+1000
0.08x+0.02y=310 Solve the system using the method of addition?=.
ok this is how my book says solve it
[2] x - y=100 -----> 2x-2y=2000
[-25] 0.08x+0.02y=310 -->-2x-0.5y=-7750
where and how did they get [-25]?
They were trying to eliminate the decimals, but they did it in a very clumsy way
I would have multiplied the second by 100
8x + 2y = 31000
then the first by 2
2x - 2y = 2000
trivial from here on ...
leaving a decimal after you multiplied by something sort of defeats the purpose, why didn't they just multiply the first by 50 and leave the first alone.
That would have done the trick also.
To solve the system of equations using the method of addition, we need to eliminate one variable by adding the two equations together.
In this case, we can eliminate the y variable since the coefficients in front of y are the same in both equations.
The original system of equations is:
1) x = y + 1000
2) 0.08x + 0.02y = 310
To eliminate y, we'll multiply equation 1 by -0.02 and equation 2 by 50, which will give us equal coefficients in front of y:
-0.02(x) = -0.02(y + 1000) (equation 1 multiplied by -0.02)
50(0.08x + 0.02y) = 50(310) (equation 2 multiplied by 50)
Simplifying these equations gives us:
-0.02x = -0.02y - 20
4x + y = 15500
Now, we can add the two equations together:
-0.02x + 4x + y = -0.02y - 20 + 15500
Simplifying further:
3.98x + y = 15480 - 0.02y
This equation can be rewritten as:
3.98x + 1y = 15480 - 0.02y
Now, we have eliminated the y variable, and the coefficient in front of y is 1. The coefficient in front of x is 3.98 (approximately), which can be rounded to 4. This is where the [-25] comes from.
So, the rewritten equation is:
4x + y = 15480 - 0.02y
The coefficient in front of y is now 1, and to balance the equation, we'll multiply both sides by -1, resulting in:
-4x - y = -15480 + 0.02y
Now, we have:
-4x - y = -0.02y - 15480
Since -0.02y is very close to -0.02y, it is rounded to -0.02y. Therefore, we get:
-4x - y = -0.02y - 15480
In summary, the [-25] is obtained by multiplying equation 2 by 50, which helps eliminate the y variable and obtain an equation with equal coefficients in front of y. The rounding of the coefficient in front of x from 3.98 to 4 results in the [-25] representation in the equation.