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Solve two unknown prices with a system of equations

Give each unknown its own variable

If two kinds of items have unknown unit prices, label them $x$ and $y$ . A total purchase gives one equation; a comparison of smaller purchases gives another. Solve the system and substitute the prices back into both original conditions.

Problem

Eight kilograms of one grade of pears and 20 kilograms of a second grade cost 6,400 rubles. Five kilograms of the first grade cost 100 rubles more than seven kilograms of the second grade. What is the price per kilogram of each grade?

Answer

First grade:

300

rubles/kg. Second grade:

200

rubles/kg.

Both purchase conditions hold for these prices.

Step-by-step solution

1. Write the system

Let $x$ and $y$ be the prices per kilogram. Then

8 x + 20 y = 6400, 5 x - 7 y = 100.

Divide the first equation by $4$ to get $2 x + 5 y = 1600$ .

2. Solve

From $2 x + 5 y = 1600$ , $x = 800 - 2.5 y$ . Substitute into the second equation:

5 (800 - 2.5 y) - 7 y = 100.

Thus $4000 - 19.5 y = 100$ , so $y = 200$ and $x = 300$ .

3. Check

$8 (300) + 20 (200) = 6400$ , and $5 (300) - 7 (200) = 100$ .

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