SOLUTION: At a local garage sale, Christ bought 4 gardening magazines and 3 mugs for $8.70; Sarah bought 5 magazines , 2 mugs and 3 refrigerator magnets for $12.35; and Steve bought 2 magazi

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Question 1120158: At a local garage sale, Christ bought 4 gardening magazines and 3 mugs for $8.70; Sarah bought 5 magazines , 2 mugs and 3 refrigerator magnets for $12.35; and Steve bought 2 magazines, 4 ceramic mugs and 4 refrigerator magnets for $15.50. What was the price of each refrigerator magnet?
Found 2 solutions by Shin123, greenestamps:
Answer by Shin123(626)   (Show Source): You can put this solution on YOUR website!
I used a solver to solve this problem, click here for the solver,
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let . This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are , , and and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get . To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: denotes the determinant of the matrix A.



---------------------------------------------------------



Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'x' column so to speak).






Now compute the determinant of to get . Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of by the determinant of to get:



So the first solution is




---------------------------------------------------------


We'll follow the same basic idea to find the other two solutions. Let's reset by letting again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'y' column in a way).






Now compute the determinant of to get .



To find the second solution, divide the determinant of by the determinant of to get:



So the second solution is




---------------------------------------------------------





Let's reset again by letting which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix






Now compute the determinant of to get .



To find the third solution, divide the determinant of by the determinant of to get:



So the third solution is




====================================================================================

Final Answer:




So the three solutions are , , and giving the ordered triple (0.75, 1.9, 1.6)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.


So each refrigerator magnet cost $1.60.
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Yes, there are ways to solve systems of equations using calculators of various sorts. But solving them with pencil and paper is a useful skill....

The statement of the problem yields three equations:
(1) 4x+3y = 8.70
(2) 5x+2y+3z = 12.35
(3) 2x+4y+4z = 15.50

Since one of the equations has only two of the variables, very probably the best way to start solving the system is to eliminate the third variable from the other two equations, yielding a system of two equation in two unknowns.

20x+8y+12z = 49.40
6x+12y+12z = 46.50
14x-4y = 2.90

Now eliminate y between this equation and equation (1).

16x+12y = 34.80
42x-12y = 8.70
58x = 43.50
x = 43.50/58 = 0.75

4(.75)+3y = 8.70
3y = 5.70
y = 1.90

2(.75)+4(1.90)+4z = 15.50
4z = 6.40
z = 1.60

The cost of each refrigerator magnet, z, was $1.60.
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