# 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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