Question 27·Hard·Linear Equations in Two Variables
A farmer bought calves and goats. He purchased 2 fewer calves than goats. Each calf eats 35 pounds of feed per week, and each goat eats 15 pounds of feed per week. During the first week, the animals consumed a total of 1,130 pounds of feed.
How many calves did the farmer purchase?
(Express the answer as an integer)
For SAT word problems with two unknown quantities, start by defining clear variables and carefully translating comparison phrases like “fewer than” or “more than” into equations that show which quantity is larger. Next, write an equation that represents the total (such as total cost, distance, or, here, total feed). Use substitution or elimination to reduce the system to a single variable, solve, and then plug back in to find the other quantity—finally double-check that you are giving the value the question actually asks for (calves vs. goats in this case).
Hints
Set up variables from the story
Let one variable stand for the number of calves and another for the number of goats. How can you write “2 fewer calves than goats” as an equation relating these two variables?
Write the total feed equation
Each calf eats 35 pounds and each goat eats 15 pounds. How can you express the total feed (1,130 pounds) in terms of the number of calves and goats?
Reduce to one variable
Use the relationship between calves and goats to substitute one variable into the total-feed equation so that you have only one variable to solve for.
Answer what the question asks
After you solve for the number of goats, use the relationship between calves and goats to find the number of calves. Be careful about whether you should add 2 or subtract 2.
Desmos Guide
Enter the equations for the situation
In Desmos, let represent the number of calves and represent the number of goats. Type the two equations:
35x + 15y = 1130(total feed)y = x + 2(2 more goats than calves, which matches “2 fewer calves than goats”).
Find and interpret the intersection
Look for the point where the two lines intersect and tap/click that point. The -coordinate of this intersection is the number of calves, and the -coordinate is the number of goats. Read off the -value for your answer.
Step-by-step Explanation
Define variables and translate the relationship
Let be the number of calves and be the number of goats.
The phrase “2 fewer calves than goats” means the number of calves is 2 less than the number of goats, so:
(Equivalently, you could write ; it represents the same relationship.)
Write an equation for the total feed
Each calf eats 35 pounds of feed per week, so all calves together eat pounds.
Each goat eats 15 pounds of feed per week, so all goats together eat pounds.
Together they eat 1,130 pounds in the first week, so:
Substitute to get an equation in one variable
Use in the feed equation .
Substitute:
Distribute and combine like terms:
50g - 70 = 1130
50g = 1200
g = 24
Use the relationship to find the number of calves
We know there are 2 fewer calves than goats, and .
So:
The farmer purchased 22 calves.