Answer:
The price of the hats is $ 10, the price of the masks is $ 5 and the price of the T-shirts is $ 15.
Explanation:
A system of equations is a set of two or more equations with several unknowns in which you want to find a common solution.
In other words, a system of linear equations is a system of equations in which each equation is linear. A solution of a system is an assignment of values for the unknowns that makes each of the equations is true, so solving a system means finding all the solutions of the system.
In this case you apply a system of equations. So you first determine the system variables:
- x: price of hats.
- y: price of the masks.
- z: price of t-shirts.
You know that Jose purchased 2 hats, 3 masks, and 1 t-shirt for $50. This is:
2*x + 3*y + 1*z=50 Equation (A)
You know that Hannah purchased 1 hat, 2 masks, and 2 t-shirts for $50. This is:
1*x + 2*y + 2*z=50 Equation (B)
You know that Amarria purchased one of each item for $30. This is:
1*x + 1*y + 1*z=30 Equation (C)
The system of equations is formed by equations (A), (B) and (C).
The method of reduction consists of adding (or subtracting) the equations to each other to eliminate one of the unknowns. Sometimes, it is necessary to multiply the equations by some number so that, when adding them, one of the unknowns disappears.
In this case you apply the method to calculate the value of z. Subtracting (A) - (B) - (C) you get:
(2-1-1)*x + (3-2-1)*y + (1-2-1)*z= 50-50-30
Solving:
0*x + 0*y + (-2)*z=-30
(-2)*z=-30
z=15
Remembering that z represents the price of the t-shirts, then each has a value of $ 15.
Replacing the value of z in equation (A):
2*x + 3*y + 1*15=50
you get:
2*x + 3*y= 50 - 15
2*x + 3*y=35 Equation (D)
Replacing the value of z in equation (B):
1*x + 2*y + 2*15=50
1*x + 2*y + 30= 50
1*x + 2*y= 50- 20
1*x + 2*y= 20 Equation (E)
The substitution method consists of choosing an equation and solving one unknown in terms of the other unknown. Then you must substitute the expression found in the other equation, to obtain an equation with an unknown, and then solve that unknown. Finally, you substitute the value found in the previously found expression to solve for the remaining unknown.
Solving for the variable x from Equation (E) you get:
x= 20- 2*y Equation (F)
Replacing in Equation (D), you get:
2*(20-2*y) + 3*y=35
Solving:
2*(20) + 2*(-2)*y + 3*y= 35
40 - 4*y + 3*y=35
- 4*y + 3*y=35 - 40
(-1)*y= -5
y=5
Remembering that y represents the price of the masks, then each has a value of $ 5.
Finally, replacing the value of y in equation (F) you get:
x=20 - 2*5
x= 20-10
x=10
Remembering that y represents the price of the hats, then each has a value of $ 10.
In summary, the price of the hats is $ 10, the price of the masks is $ 5 and the price of the T-shirts is $ 15.