Linear equations

Summary and examples

An example of a linear equation is:

2(x+3)-3x+2=3(x-2)+2(x+1)+6

and the question is to calculate the value of x that makes the equation an identity. We call this solving the equation.

To solve this equation, we proceed as follows.

1. First, we remove the parentheses:

2x+6-3x+2=3x-6+2x+2+6

2. Then, in the left- and right-hand side we gather the x‘s and constants:

-x+8=5x+2

3. Now we bring the x‘s to the left-hand side and the constants to the right-hand side:

-x-5x=2-8

resulting in:

-6x=-6

4. We divide both sides of the equation by the coefficient (\neq0) of x, in this case by -6. Then we get:

x=1

which is the solution of the equation.

It is possible that this procedure leads to an equation of the form:

0.x=c

Then there are two possibilities.

1. c\neq0

In this case no value of x can satisfy the equation. The left-hand side always equals 0, while the right-hand side is not equal to 0. The equation has no solution.

2. c=0

Now we have the equation:

0.x=0

and any value of x satisfies the equation.

Example 1

Solve:

2x+3=5x-8

We can write this equation as:

2x-5x=-8-3

so:

-3x=-11

and thus:

x=\displaystyle\frac{11}{3}

Example 2

3(x^2+3x-5)=x^2+x+2(x^2-x)+5

We can write this equation as:

3x^2+9x-15=x^2+x+2x^2-2x+5

or:

9x-15=x-2x+5

or:

10x=20

and thus:

x=2

Note that at first sight the equation seems a quadratic equation. However, the squares of x disappear, with a linear equation as a result.

Example 3

Solve:

2x+3x+5(2x-3)=30

We can write this equation as:

5x+10x-15=30

or:

15x=45

and thus is the solution:

x=3

Example 4

Solve:

3x+2(x-5)=5x+2

We can write this equation as:

3x+2x-10=5x+2

or:

0.x=12

No value of x can satisfy this equation and thus the equation has no solution.

Example 5

Solve:

10(x-2)-4(2x-1)=2(x-1)-14

We can write this equation as:

10x-20-8x+4=2x-2-14

or:

0.x-16=-16

or:

0.x=0

Any value of x satisfies this equation. This equation has an infinite number of solutions.

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