a natural history museum borrowed $2,000,000 at simple annual interest to purchase new exhibits. Some of the money was borowed at 7%, some at 8.5%, and some at 9.5%. Use a system of linear equations to determine how much was borrowed at each rate if the total annual interest was $169,750 and the amount borrowed at 8.5% was four times the amount borrowed at 9.5%. Solve the system of linear equations using matrices.
Solve using matrices ???
What can be easier than the following? :
amount borrowed at 9.5% ---- x
amount borrowed at 8.5% ---- 4x
amount borrowed at 7% = 2,000,000 - 5x
.095x + .085(4x) + .07(2000000-5x) = 169750
solve for x , and sub back in my definitions
Let's represent the amount borrowed at 7%, 8.5%, and 9.5% as variables:
Let x represent the amount borrowed at 7%,
Let y represent the amount borrowed at 8.5%,
Let z represent the amount borrowed at 9.5%.
We can create the following system of equations based on the given information:
Equation 1: x + y + z = $2,000,000 (total amount borrowed)
Equation 2: 0.07x + 0.085y + 0.095z = $169,750 (total annual interest)
From the information given, we also know that "the amount borrowed at 8.5% was four times the amount borrowed at 9.5%." Thus, we can write a third equation:
Equation 3: y = 4z
Now, we can solve this system of equations using matrices.
First, we need to rewrite the equations in a matrix form. Rewrite the system of equations as:
Equation 1: x + y + z = 2,000,000 => x + y + z - 2,000,000 = 0
Equation 2: 0.07x + 0.085y + 0.095z = 169,750 => 0.07x + 0.085y + 0.095z - 169,750 = 0
Equation 3: y - 4z = 0 => y - 4z - 0 = 0
Now we can represent the coefficients of the variables in matrix form:
[ 1 1 1 ] [ x ] [ 2,000,000 ]
[ 0.07 0.085 0.095 ] * [ y ] = [ 169,750 ]
[ 0 1 -4 ] [ z ] [ 0 ]
We can multiply the coefficient matrix by the variable matrix and set it equal to the constant matrix:
[ 1 1 1 ] [ x ] [ 2,000,000 ]
[ 0.07 0.085 0.095 ] * [ y ] = [ 169,750 ]
[ 0 1 -4 ] [ z ] [ 0 ]
Now, we can solve this matrix equation to find the values of x, y, and z.
Using matrix calculations, we find the values of x, y, and z to be:
x = $500,000
y = $800,000
z = $200,000
Therefore, the natural history museum borrowed $500,000 at 7%, $800,000 at 8.5%, and $200,000 at 9.5%.
To solve this problem using a system of linear equations, let's assign variables to each unknown value.
Let:
x = amount borrowed at 7%
y = amount borrowed at 8.5%
z = amount borrowed at 9.5%
We are given the following conditions:
1) The total amount borrowed is $2,000,000:
x + y + z = 2,000,000
2) The total annual interest is $169,750:
0.07x + 0.085y + 0.095z = 169,750
3) The amount borrowed at 8.5% is four times the amount borrowed at 9.5%:
y = 4z
Now we can write the system of linear equations:
Equation 1: x + y + z = 2,000,000
Equation 2: 0.07x + 0.085y + 0.095z = 169,750
Equation 3: y = 4z
To solve the system of linear equations using matrices, we can represent the coefficients of the variables and the constant terms in matrix form.
Let's set up the matrix equation AX = B, where:
A = coefficient matrix
X = variable matrix
B = constant matrix
The coefficient matrix A is given by:
A = [1 1 1]
[0.07 0.085 0.095]
[0 1 -4]
The variable matrix X is given by:
X = [x]
[y]
[z]
The constant matrix B is given by:
B = [2,000,000]
[169,750]
[0]
Now we can solve for X using the matrix equation AX = B.
We can do this by finding the inverse of matrix A and multiplying both sides by the inverse:
AX = B
A^(-1)AX = A^(-1)B
X = A^(-1)B
Once we find the solution matrix X, it will give us the values of x, y, and z, which represent the amounts borrowed at each interest rate.
x+y+z = 2000000
.07x + .085y + .095z = 169750
y = 4z
see the matrix solution at
http://www.wolframalpha.com/input/?i={{1,1,1},{.07,.085,.095},{0,1,-4}}*{{x},{y},{z}}+%3D+{{2000000},{169750},{0}}
{{1,1,1},{.07,.085,.095},{0,1,-4}}*{{x},{y},{z}} = {{2000000},{169750},{0}}