Solving Simple Linear Equation In One Variable

Before you got to know the concept of equations, you used them in your daily life without realizing it. do you believe that?

Yes, you used mental calculation to solve simple equations, perhaps before you entered school. For example, if I tell you what number if we add it to 8 we get 10. Here you do a mental calculation that includes addition and subtraction, but in a very simple way only, where you subtract 8 from 10 and you get 2, and to make sure of your solution you add the number 2 to 8 and you get 10. That's how simply all the children passed this stage.

The same thing if I told you what number that if we multiply it by 5 we get 20. You do a simple mental calculation that includes multiplication and division, where you divide 20 by 5 to get 4, and to make sure of your answer multiply 4 by 5 to get 20. This is what we will simply rely on in this lesson to solve very simple linear equations. Continue reading the article until the end if you are interested in this lesson.

The concept of an equation

An Equation is a mathematical expression consisting of one or more unknown values, and a right and a left side. These two sides are always separated by the symbol.

A Linear Equation is an equation with one unknown. This unknown can be in one or several terms.

Examples of Linear Equations:

The root of the Equation is the value of the unknown number that we get after solving the equation. When we plug this number back into the equation, we should get both sides equal.

Standard Form Of A Linear Equation In One Variable

Note:

Let and represent two real numbers with

The standard form of a linear equation with one unknown is:

The solution of this equation is

In addition to linear equations, there are other types of equations, which we will get acquainted with next time:

• Linear Equations with two unknowns;

• Quadratic equations(one unknown with second degree);

• Rational equations

• Cubic equations

• etc.

How To Solve Simple Linear Equation

To solve an equation with one unknown, the unknown must be isolated in the equation by following these four steps in this order.

• Transfer Rule: We shift the unknowns to the left side and shift the known numbers to the right side by changing the sign of the transferred terms;

• Simplify the expressions in the two sides obtained in the previous step;

• In this step, we multiply both sides by the reciprocal of the unknown coefficient;

• Simplify the expressions in the two sides;

Example1:

Solve the equation:

Solution:

The unknown of this equation is , and its coefficient is 4.

• We must transfer -1 from the left side to the right side and changing the sign.
• Simplify the expression obtained.
• We must myltiply the both sides by the reciprocal of the coefficient of . i.e, we multiply by
• Simplify the expression to isolate .
• The final answer is . We substitute the value of by 2 into the equation to make sure the answer is correct.

Example2:

Solve the equation:

Solution:

The unknown of this equation is , and its coefficient is -3.

• We must transfer 5 from the left side to the right side and changing the sign.
• Simplify the expression obtained.
• We must myltiply the both sides by the reciprocal of the coefficient of . i.e, we multiply by
• Simplify the expression to isolate .
• The final answer is . We substitute the value of by 5 into the equation to make sure the answer is correct.

Example3:

Solve the equation:

Solution:

In this expression, we have the unknown x found on both sides with different coefficients.

• We must transfer 5 from the left side to the right side and, transfer from the right side to the left side and changing the signs.
• Simplify the expression obtained. In this case we simplify the left side by Combining Like Terms.
• We must myltiply the both sides by the reciprocal of the coefficient of . i.e, we multiply by
• Simplify the expression to isolate .
• The final answer is . We substitute the value of by into the equation to make sure the answer is correct.

Example4:

Solve the equation:

Solution:

This equation is also a linear equation, but it contains expressions in parentheses. So it is necessary to first get rid of these parentheses using the distributive property.

• Applying distributive property;
• We must transfer 17 from the left side to the right side and, transfer from the right side to the left side and changing the signs.
• We must myltiply the both sides by the reciprocal of the coefficient of . i.e, we multiply by
• Simplify the expression to isolate .
• The final answer is . We substitute the value of by 3 into the equation to make sure the answer is correct.

Example5:

Solve the equation:

Solution:

The unknown of this equation is , and its coefficient is .

• We must transfer -1 from the left side to the right side and changing the sign.
• Simplify the expression obtained.
• We must myltiply the both sides by the reciprocal of the coefficient of . i.e, we multiply by
• Simplify the expression to isolate .
• The final answer is . We substitute the value of by 3 into the equation to make sure the answer is correct.

Example6:

Solve the equation:

Solution:

The best way to solve such equations in my opinion is to make all terms have the same denominator. In order to reduce it and then make it easier for us to solve the equation.

Learn how to add and subtract two fractions with:

The same denominator is the Greatest Common Factor of all denominators in the equation. In our case, We must find the GCF of 3 and 4. It's clear that , so:

After that, we must applying the distributive property.

Then, apllying the transfer rule.

And multiplying the both sides by .