Final answer:
To obtain the desired 291 gallons of milk containing 6% butterfat, a system of equations can be used to find the amounts of milk with different butterfat percentages that need to be used.
Step-by-step explanation:
To find out how many gallons of milk each of different butterfat percentages must be used to obtain the desired 291 gallons of milk containing 6% butterfat, we can use a system of equations.
Let's assume x gallons of milk containing 7% butterfat will be used, and y gallons of milk containing 4% butterfat will be used.
The total gallons of milk can be expressed as: x + y = 291
The total butterfat in the milk can be expressed as: 0.07x + 0.04y = 0.06(291)
Solving this system of equations will give us the values of x and y, which represent the gallons of each type of milk that need to be used to obtain the desired milk.
Method 1: Substitution
Solve the first equation for y: y = 291 - x
Substitute this expression for y in the second equation: 0.07x + 0.04(291 - x) = 0.06 * 291
Simplify and solve for x: 0.03x = 9.66, x = 322 gallons
Substitute x back into the first equation to find y: 322 + y = 291, y = -31 gallons (This negative value doesn't make sense in the context of the problem, so we need to revise our approach.)
Method 2: Elimination
Multiply the first equation by 4: 4x + 4y = 1164
Multiply the second equation by 7: 0.49x + 0.28y = 165.7
Subtract the second equation from the first equation: 3.72x = 998.3
Solve for x: x = 268 gallons
Substitute x back into the first equation to find y: 268 + y = 291, y = 23 gallons
Therefore, the dairy needs 268 gallons of milk containing 7% butterfat and 23 gallons of milk containing 4% butterfat to obtain the desired 291 gallons with 6% butterfat.