# Linear Equations

A linear equation or first-degree in $x$ is written in standard form as

with $a \ne 0$

This is solved as follows:
$ax + b = 0$
$ax = - b$   (we subtracted $b$ from both sides)
$x = - \frac{b}{a}$     (we divided both sides by $a$)

In many cases, simple first-degree equations can be solved mentally.

For example, the solution of $5x = 10$  is $x = 2$ and the solution of $2x + 3 = 0$ is $x = - \frac{3}{2}$

Example:
Solve the linear equation $29 - 2x = 15x - 5$

Solution:
We have
$29 - 2x = 15x - 5$
$29 - 2x - 15x = - 5$ (we subtracted $15x$ from both sides)
$29 - 17x = - 5$ (we combined like terms)
$- 17x = - 34$ (we subtracted $29$ from both sides)
$17x = 34$ (we multiplied both sides by $- 1$)
$x = \frac{{34}}{{17}}$ (we divided both sides by $17$)
$x = 2$

To guard against errors in arithmetics or algebra, it’s a good idea to check the solution by substituting it back into the original equation. Thus, if we substitute $x = 2$ in the equation $29 - 2x = 15x - 5$, we obtain $29 - 2\left( 2 \right) = 15\left( 2 \right) - 5$, $25 = 25$. This shows that $2$ is indeed the solution.

Example:
Solve the linear equation

Solution:

Multiplying both sides of the equation by the LCD $y\left( {y - 1} \right)$ and simplifying, we have

That is, $1 - \left( {y - 1} \right) = y$   or   $2 - y = y$

Adding $y$ to both sides of the last equation, we obtain
$2 = 2y$;    that is    $2y = 2$

From which it follows that $y = 1$. We now check by substituting $y = 1$ into the original equation to obtain

an equation in which neither side is defined because of the zeros in the denominators. In other words, the substitution $y = 1$ doesn’t make the equation true, it makes the equation meaningless. Here, the correct conclusion is that the original equation has no root.