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Determine whether the ordered triple is a solution to the system.
We will check each equation by substituting in the values of the ordered triple for and
The ordered triple is indeed a solution to the system.
Given a linear system of three equations, solve for three unknowns.
Find a solution to the following system:
There will always be several choices as to where to begin, but the most obvious first step here is to eliminate by adding equations (1) and (2).
The second step is multiplying equation (1) by and adding the result to equation (3). These two steps will eliminate the variable
In equations (4) and (5), we have created a new two-by-two system. We can solve for by adding the two equations.
Choosing one equation from each new system, we obtain the upper triangular form:
Next, we back-substitute into equation (4) and solve for
Finally, we can back-substitute and into equation (1). This will yield the solution for
The solution is the ordered triple See [link] .
In the problem posed at the beginning of the section, John invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. John invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670. How much did he invest in each type of fund?
To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts:
The first equation indicates that the sum of the three principal amounts is $12,000.
We form the second equation according to the information that John invested $4,000 more in mutual funds than he invested in municipal bonds.
The third equation shows that the total amount of interest earned from each fund equals $670.
Then, we write the three equations as a system.
To make the calculations simpler, we can multiply the third equation by 100. Thus,
Step 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up.
Step 2. Multiply equation (1) by and add to equation (2). Write the result as row 2.
Step 3. Add equation (2) to equation (3) and write the result as equation (3).
Step 4. Solve for in equation (3). Back-substitute that value in equation (2) and solve for Then, back-substitute the values for and into equation (1) and solve for
John invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds.
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