Resolved Problems of Systems of Linear Equations

• Introduction

• 16 Resolved Problems (two equations and two unknown factors)

Introduction

A system of equations is a group of equations (in our case it will be two) and various unknown factors (two in our case) that appear in one or various of these equations.

An equation with more than one unknown factor informs of the relation between them. For example, the equation x - y = 0 tells that x and y are the same number.

We cannot solve an equation with two unknown factors, because one of them is dependent on the other. For instance, if we have the equation x - 2y = 0 and we isolate x, we obtain that x = 2y. So, the value of x is double the value of y. But we still do not know the values of x and y.

To resolve a system of N unknown factors, we need N equations. In fact, we need the equations to be linearly independent, but we will not take that into consideration at this level.

In this section we have exercises that require to be considered as a system of equations of second dimension (two equations and two unknown factors). If we do not remember how to resolve systems (equalization, reduction and substitution), we can visit the section of Systems.

Resolved Problems

Problem 1

Two numbers add 25 and the double of one of them is 14. What numbers are they?

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Problem 2

The double of the sum of the two numbers is 32 and the subtraction of them is 0. What numbers are they? We apply reduction.

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Problem 3

Adding two numbers gives us 12 and half of one of them is double the other. What numbers are they?

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Problem 4

We have two numbers that add 0, and if we add 123 to one of them, we get the double of the other one. What numbers are they?

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Problem 5

Find the two digit number so that: the second digit is double the first and adding both digits gives us 12.

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Problem 6

Anne is three times the age of her son Jacob. In 15 years, Anne's age will be double her son's. How many years older is Anne than Jacob?

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Problem 7

We have bought 3 glass marbles and 2 steel ones for 1,45$and, yesterday, 2 glass ones and 5 steel ones for 1,7$. Determine the price of one glass marble and a steel one.

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Problem 8

Work out the length of the sides of a rectangle that its perimeter is 24 and its large side measures three times the short side.

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Problem 9

Work out the number of animals in a farm knowing that:

• the sum of ducks and cows is 132 and their legs add 402.

• 200 Kg are needed a day to feed the chickens and the roosters. There is one rooster per 6 chickens and the average amount a chicken eats is 500g, double what the rooster does.

• It is known that the sixth part of the rabbits escapes to the cows keep, which means there is triple the animals in the keep.

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Problem 10

In a multiple choice exam, the correct questions add a point and the wrong answers subtract half a point. In total there are 100 questions and none can be left unanswered (all must be answered). The student gets 8.05 out of 10 in the exam. Work out the number of questions answered correctly and incorrectly.

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Problem 11

If we add 7 to the numerator and denominator of a fraction, we obtain this fraction

$$\frac{2}{3}$$

If instead of adding 7 we subtract 3 from the numerator and denominator, we obtain this fraction

$$\frac{1}{4}$$

Find said fraction.

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Problem 12

A make of soft drinks makes a lemonade (water with lemon concentrate) with a very specific quantity of ingredients. The relation between the quantity of water and concentrate is

$$L_L = \frac{2 L_A}{5}$$

where LL represents the litres of concentrate and LA the litres of water.

If 20 lemons are needed to make one litre of lemon concentrate, how many lemons will they need to make 1230 2L bottles of this lemonade?

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Problem 13

High difficulty

With a 34 metre rope we can draw a rectangle (without there being any rope left over) that has a diagonal of 13 metres.

Calculate how much the base and height of the rectangle are.

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Problem 14

In a charity concert they sell all the tickets and the raise 23 thousand dollars. The price of the tickets is 50 for normal ones and 300 dollars for the V.I.P.

Work out the number of tickets of each type that were sold if the venue holds 160 people.

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Problem 15

A child makes an observation in the ball play park:

• There are green, red and yellow balls.

• The number of green balls and red balls is five times the yellow ones.

• The number of green ones are three times the yellow ones.

• The total amount of yellow and red balls is 123.

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Problem 16

Calculate the number of positive numbers with three digits (bigger than 99) so that one of the digits is 0 and the other two add 7.

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