How to solve linear equations in one variable

A solution of an arbitrary linear equation in one variable cannot be given as a formula. But there are four steps which will guide you to finding a/the solution(s).

To complete the four steps successfully, you need a good understanding of

1. PEMDAS - the order of operations as indicated by the letters of this word: parentheses, exponential expressions, multiplication, ... .

2. Like-terms and unlike-terms - like-terms have the same variable(s), each with the same power.

3. Variable terms and how to collect and combine like-variable-terms.

4. Constant terms and how to collect and combine constant terms

5. Inverse operations - two operations that if combined cancel each other. Addition and subtraction are inverse to each other and so are multiplication and division.

Here are the four steps:

1. Simplify the expression on each side of the equality sign, using PEMDAS. If there are like-terms in the expression on either side of the equality sign collect and combine them in on each side.

2. If there are like-terms on opposite sides of the equality sign collect and combine them in such a manner that variable term is on the left-hand-side of the equality sign and the constant term is on the right-hand-side.

3. If the variable term has a constant factor not equal to 1, divide the variable term and the constant term on the right-hand side of the equality sign by this factor. You have got the solution now.

4. Check the solution. If you replace the variable by the solution in the original equation it must elvolve to a true expression.

Here are three examples which will also introduce you to the three types of linear equations in one variable.

1. $3 \cdot (x + 2) - 4 x = 2 x - 10 + 5 x$

Step 1: PEMDAS, distributive law: $3 x + 6 - 4 x = 2 x - 10 + 5 x$

Collect and combine like-terms on each side: $- x + 6 = 7 x - 10$

Step 2: Collect and combine like-terms on opposite sides, -7x; -6: $- x - 7 x = - 10 - 6$

$- 8 x = - 16$

Step 3: Divide by (-8): $x = 2$ The solution is 2.

Step 4: Check the solution: $3 \cdot (2 + 2) - 4 \cdot 2 = 2 \cdot 2 - 10 + 5 x$

$3 \cdot 4 - 8 = 4 - 10 + 10$

$4 = 4$ This is a true expression, so the solution is 2.

If there is only a single value that satisfies the original equation, the linear equation is called conditional.

2. $3 \cdot (2 x - 1) = 7 + x - 10 + 5 x$

Step 1: PEMDAS, distributive law: $6 x - 3 = 7 + x - 10 + 5 x$

Collect and combine like-terms on each side: $6 x - 3 = 6 x - 3$

Step 2: Collect and combine like-terms on opposite sides, -6x; +3: 0 = 0

Step 3: Not applicable as there is no variable-term.

The above linear equation is satisfied by any real number, it has an infinite number of solutions. It is called an identity.

Step 4: Test your result by choosing a random number, for example, -1.

$3 \cdot (2 \cdot (- 1) - 1) = 7 - 1 + 10 + 5 \cdot (- 1)$

$- 9 = - 9$ This is a true expression. Any real number is a solution to the above equation.

$3 \cdot (2 x - 1) = 7 + x - 9 + 5 x$

Step 1: PEMDAS, distributive law: $6 x - 3 = 7 + x - 9 + 5 x$

Collect and combine like-terms on each side: $6 x - 3 = 6 x - 2$

Step 2: Collect and combine like-terms on opposite sides, -6x; +3: $0 = - 5$

Step 3: Not applicable as there is no variable-term.

The above linear equation has no solution. It is called a contradiction.

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