To solve this problem, let's work through the given equations step-by-step.
We start with two equations:
1. [tex]\( x + y = 8000 \)[/tex] \\
(This represents the total amount of money invested.)
2. [tex]\( 0.10x + 0.12y = 900 \)[/tex] \\
(This represents the total yearly interest from both investments.)
First, let's rewrite these equations for clarity:
1. [tex]\( x + y = 8000 \)[/tex] \\
2. [tex]\( 0.10x + 0.12y = 900 \)[/tex]
We need to solve these equations simultaneously to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
From the first equation, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 8000 - x \][/tex]
Next, we substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ 0.10x + 0.12(8000 - x) = 900 \][/tex]
Now, expand and simplify the equation:
[tex]\[ 0.10x + 0.12(8000) - 0.12x = 900 \][/tex]
[tex]\[ 0.10x + 960 - 0.12x = 900 \][/tex]
Combine like terms:
[tex]\[ -0.02x + 960 = 900 \][/tex]
Subtract 960 from both sides of the equation:
[tex]\[ -0.02x = 900 - 960 \][/tex]
[tex]\[ -0.02x = -60 \][/tex]
Divide both sides by [tex]\(-0.02\)[/tex]:
[tex]\[ x = \frac{-60}{-0.02} \][/tex]
[tex]\[ x = 3000 \][/tex]
So, the amount invested at [tex]\( 10\% \)[/tex] is [tex]\( \$ 3000 \)[/tex].
Now we can find [tex]\( y \)[/tex] by substituting [tex]\( x = 3000 \)[/tex] back into the first equation:
[tex]\[ y = 8000 - 3000 \][/tex]
[tex]\[ y = 5000 \][/tex]
This means the amount invested at [tex]\( 12\% \)[/tex] is [tex]\( \$ 5000 \)[/tex].
Therefore, the total amount of [tex]\( \$ 8000 \)[/tex] was divided such that [tex]\( \$ 5000 \)[/tex] was invested at [tex]\( 12\% \)[/tex] interest.
