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The course on equations **of the first degree with one unknown** with the definition and the properties of solving an equation and then the method of solving problems in the fourth grade (4e) is to be understood completely. The student should be able to solve an equation with the various properties of the course with transposition of terms and factors in each of the members.

We will end this lesson by trying to solve problems from everyday life in the fourth grade.

## I. First degree equations with one unknown

An equation of the first degree with one unknown is any equality that can be reduced to

to the following form: ax + b = c (with a, b and c three given relative numbers).

- x is called the unknown of this equation.
- the expression to the left of the = sign is called the “first member” of the equation.
- the expression to the right of the = sign is called the “second member” of the equation.
- Solving an equation means finding all values of x such that the first member is equal to the second member.

Examples:

Here are some equations but they are not necessarily of the first degree with one unknown.

- The solutions of an equation are not changed if the same quantity is added (or subtracted) to each member of the equation.
- The solutions of an equation are not changed by multiplying (or dividing) each member of the equation by a non-zero relative number.

Example:

Solve the equation .

The solution of this equation is .

## II. Solving first degree problems with one unknown

Here are the different steps to solve first degree problems with one unknown :

**Methodology:**

1) Choice of the unknown

2) Putting it into an equation

3) Solving the equation

4) Verification

5) Conclusion

**Applications:**

**Problem 1:**

A two-pan balance is in equilibrium when 10 cubes and a mass of 2 kg are placed on one pan and 2 cubes and a mass of 30 kg on the other. What is the mass of a cube?

**Problem 2:**

A father is 30 years old and his son is 10 years old.

In how many years will the father’s age be double that of the son?

**Problem 3:**

Bleach is used diluted. A 2% diluted solution contains 2 cl of bleach and 98 cl of water to form one liter (100 cl) of solution. A 30% diluted solution contains 30 cl of bleach to form one liter (100 cl) of solution.

How much 30% solution must be added to 1 liter of a 2% solution to make a 10% solution?

**Problem 4:**

A 10 m high tree and a 2 m high pole are located opposite each other on each bank of a 30 m wide river. At the top of each of them is perched a bird. They both launch themselves at the same speed and at the same time on a poor fly that taunts them on the surface of the water. By a magical effect of Mother Nature, they reach it at the same time and smash their beaks in a more than vigorous contact, and thus find themselves empty-handed.

How far from the foot of the tree was this miraculous fly?

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