Before explaining how to solve equations, it is necessary that you know what an equation is: it is an equality that contains unknowns, terms and members. Let us define each of these words, so that you do not remain with the idea in the inkwell:
First, an equality is an expression that contains the equals symbol (=). Equalities can be numeric and literal or algebraic. A numerical equality is one that contains only numbers, for example: 2+3=5. Literal or algebraic equalities are those that contain numbers and letters. These in turn can be of two types: identities or equations.
Identities are those true for any value that is given to the unknowns, example: On the contrary, an equation is an equality that is only true for some values of the unknown or variables.
An unknown, also called a variable, is a letter whose value is unknown. It should be noted that the letters used as variables most frequently are x, y, wyz; however, any letter is perfectly usable as a variable or unknown.
The terms are each one of the parts of the equation separated by the symbols of + or -.
And finally, the members are each of the expressions that are on each side of the equation, thus identifying two members: the first member, which is what is on the left side of the equation, and the second term, which is the expression which is on the right side of it.
Parts of an equation
Look at the following example in which each of the parts of an equation are identified:
Importance and function of the equations
The equations are of great importance and use in everyday life; although many times we do not realize it and it is not necessary to raise the problem formally.
For example, if we are going to buy a smartphone in our favorite virtual store; and it costs €500, and we have €400, it is easy to know that we would lack $100 to buy the product. But, let’s see how to write it in the form of an equation: x+400€=500€
In this case, we know that x=100, because 100+400=500, but in the case of equations that are not so easy to solve, how would we proceed?
Instructions for solving equations
Solving an equation is finding its solution, that is, finding the value of the unknown. For this, it is necessary to know the properties of equalities, namely:
- If a number is added or subtracted from both sides of the equality, the equality is not altered.
- If a number is multiplied or divided on both sides of the equality, this does not change.
The equations are classified according to the exponent of the variable. In this section, we will study first degree equations with one unknown; Also, quadratic equations with one unknown.
In order to solve the first and second degree equations with an unknown, it is necessary to take into account the properties. Both of addition and multiplication in sets, namely: associative property, commutative property, neutral element, opposite element, zero factor and the distributive property of the product with respect to addition.
You also need to know the definition of like terms. Which are those that have the same variable raised to the same exponent; and the sign rules for addition and multiplication.
Finally, you need to know the order of operations. This is (if there are no grouping signs, be it parentheses, brackets and/or braces): powers are solved first, then multiplications and divisions and then additions and subtractions.
Solved exercises
First degree equations with an unknown, are those where the variable has an exponent 1 (which is understood).
Let’s see some exercises:
X+2=5
- To solve an equation it is necessary to isolate x. This means “leave it alone” in one of the members of equality; it is usually left alone on the left side.
- In this case, to achieve this it is necessary to remove the +2.
- To do this, it is necessary to subtract 2. This so that when adding or subtracting it results in zero, remembering the property of the neutral element for addition: any number added with zero results in the same number.
- Thus, when subtracting 2 from the first member of the equality, which is where the variable is located, in this case x, it is necessary to also subtract 2 from the other member of the equality so that it does not change (recalling the properties of inequalities ).
Let ‘s see the procedure followed to solve this equation:
As you noticed, the variable has a value of 3. To know if that result is correct, it is necessary to check the equation. Checking an equation is nothing more than checking if its result is correct or not. To do this, the following steps must be followed:
- The letter is replaced or changed by the result.
- The operations on each side of the equality are solved WITHOUT CHANGING ANY TERM.
- If the same value is observed on each member of the equality, then the result of the equation is correct, otherwise it is not.
Let ‘s see how to check this exercise:
To solve this exercise: 3+y=7, proceed in a similar way as with the previous exercise, regardless of whether the variable, in this case, is at the beginning of the equation or not:
Verification:
In this case, let’s see that on the side of the variable or unknown, not only do we have to eliminate the number that remains, in this case 2, but there is also a number that multiplies.
To do this, one can proceed by eliminating first any of the two numbers, and then the next one, paying attention again to the properties of the equalities.
Note that when there is no sign between a number and the variable or unknown, it is understood that you are multiplying the variable. Generally and to facilitate the calculations, the number that is adding or subtracting is eliminated first, in this case, subtracting, like this:
As can be seen, the variable z has not yet been solved. To know its value, it is necessary to eliminate the 2 that is multiplying it. To do this, it is divided on both sides by the number 2 so that the equality is not altered, this in order that when dividing a number by it, the result is 1 and, applying the property of the neutral element for multiplication: all number multiplied by 1 gives the same number, so:
Notice that in this case the result is a fraction. You can give the result as a fraction or as a decimal. In this case, the decimal number resulting from this fraction is 7.5. Now, we will use fractions as solutions.
In this case it is necessary to first solve the operation on the right hand side of the equality in order to simplify operations, like this:
w/5+1=31+5
w/5+1=36
Then, proceed as in the previous exercise, with the difference that, in this case, both members of the equality must be multiplied by 5 so that the equality does not change.
Verification:
To solve this exercise: 2x+4=3x+1, as the variable appears more than once, we recommend that you first place the terms that contain the variable on one member, and the terms that do not contain on the other member. the variable, that is, group like terms, like this:
Now, let’s proceed to solve the operations on both sides. Since 2x and 3x are similar terms, we proceed to operate normally, that is, the subtraction is solved, attending to the rule of signs for addition and the x is placed, in this way:
2x-3x=1-4
-1x=-3
Then, divide both sides of the equation by -1. So, remembering the rules of signs for multiplication:
What do you need to solve equations?
To solve quadratic equations with an unknown, that is, equations where the variable is raised to the exponent 2, it is necessary to know the formula of the solver, which is given by:
x=(-b±√(b^2-4ac))/2a
Notice that the symbol appears in the formula. This means that there will be a result for when it is solved with + and another for when it is solved with -. When √(b^2-4ac)=0 both results will be equal, that is, the equation will have only one solution.
The x is the variable to be cleared, which in this case is raised to the square, and the letters a, b and c, are the coefficients of the variable and the independent term. Thus, this formula is applicable to isolate variables in exercises of the form: ax 2 +bx+c=0.
Which means that it is enough to reduce the given equations to an expression similar to this one in order to apply the solvent formula and obtain the value of x.
In the cases of quadratic equations, x can have 1 solution, it can have 2 solutions, and it can have no solution.
Tips for Solving Equations
To solve equations of the first and second degree with an unknown in a faster and easier way, the concept of equivalent equations can be considered: two equations are equivalent if the same result is obtained when solving them. Taking this definition and trying to summarize the properties of inequalities, the following “rules” could be derived:
- What is adding in one member of equality “passes” to the other member by subtracting.
- What is subtracting in a member of equality “passes” to the other member by adding.
- What is multiplying in one member of equality “passes” to the other member by dividing.
- What is dividing in one member of the equality “passes” to the other member multiplying.
In this way, let us solve the following exercise again, taking into account these rules:
Note that the same result was obtained, summing up the number of steps.
A simpler way to solve some quadratic equations is taking into account the coefficients of the variable, when it is already operated to be able to apply the formula of the solver, that is, in the form: ax 2 +bx+c=0
If a=0, then we could look for 2 numbers that multiply to give c and add to give b. It is necessary to take into account that these numbers are not always easy to find by mere inspection. Then, the resolvent formula would already be applied as we saw previously.
Let’s see if the exercise that we solved of second degree equations can be solved in this way:
Two numbers that add up to -6, in this case they would be -3 and +2 or +3 and -2. Now, two numbers that multiply to 1, as it is positive and we must choose one of the two options given, it would be +3 and -.2, since 3-2=1. Then we have the following: (x+3)∙(x-2)=0
Then both numbers equal 0 because, according to the Zero Factor Property for Multiplication, one of the 2 factors must be 0 for the equality to be true. You will notice that it turned out the same as applying the resolvent.