4. Translate the problem into a pair of linear equations in two vaiables. Solve the equations using either elimination or substitution. State your answer for the specified variable.
Don runs a charity fruit sale, selling boxes of oranges for $11 and boxes of grapefruit for $10. If he sold a total of 762 boxes and took in $8125 in all, then how many boxes of oranges did he sell?
11Oranges+ 10G= 8125
G+ Oranges= 762
To solve this problem, we need to translate the information given into a pair of linear equations in two variables. Let's assign variables to represent the number of boxes of oranges and grapefruits Don sold.
Let x be the number of boxes of oranges.
Let y be the number of boxes of grapefruits.
From the problem, we can deduce two equations:
1. The total number of boxes sold: x + y = 762
2. The total amount of money earned: 11x + 10y = 8125
Now that we have the equations, we can solve them using either elimination or substitution. Let's use the elimination method.
To eliminate one of the variables, we can multiply the first equation by a suitable number to make the coefficients of one of the variables equal. Since the coefficient of y in the first equation is already 1, we'll multiply the second equation by 1 to make the coefficients of y equal:
11(x + y) = 11(762)
11x + 11y = 7621
Now we have the system of equations:
11x + 11y = 7621 ---(3)
11x + 10y = 8125 ---(4)
To eliminate x, we can subtract equation (4) from equation (3):
(11x + 11y) - (11x + 10y) = 7621 - 8125
11y - 10y = -503
Simplifying, we get:
y = -503
Now let's substitute the value of y back into one of the original equations. We'll use equation (1):
x + (-503) = 762
x - 503 = 762
x = 762 + 503
x = 1265
Therefore, Don sold 1265 boxes of oranges.