V Equations

Solving equations

Since both sides of an equation are either equal (true equation) or unequal (untrue equation), performing the same operation on both sides yields in the same answer. Multiplication and division have to be applied to all terms of an expression, while additions and subtractions are only applied once per expression. Examples:
$x - 4 = -1$ | $+4$ (isolate x to find its value)
$x = 3$

When a variable shows up in both expressions, elimite the smaller one through subtraction:
$9x = 8x + 4$ | $-8x$
$9x - 8x = 4$
$x = 4$ $(1x = x)$

Because $2x = 2 \times x$, division has to be used to solve equations like $2x = 10$:
$2x = 10$ | $\div2$
$x = 5$

If the coefficient is negative the expressions should be divided by a negative number:
$-7y = 35 \Harr (-7)y = 35$ | $\div-7$
$y = -5$

Considering that fractions signify division we multiply by the denominator:
$\frac x 5 = -3$ | $\times35$ $x = -3 \times 5$
$x = -15$

$-x = -4$ | $\div-1$
$x = 4$

Simplifying Expressions

Elimination of Fractions in Equations

Fractions in equations can be removed by following these steps: $y - \frac 2 3 = \frac {5}{12}$

1. Write everything as a fraction. Terms that are not fractions become numerators over 1 ($x = \frac x 1$).
   $\frac y 1 - \frac 2 3 = \frac {5}{12}$
   
2. Find the lowest common denominator (12)
3. Multiply every term by the LCD ($\frac {12 \times y}{1 \times 1} - \frac {12 \times 2}{1 \times 3} = \frac {12 \times 5}{1 \times 12}$)
4. Simplify to remove all fractions ($12y - 8 = 5$)

If a fraction serves as coefficient to a variable, the variable is moved to the numerator of the coefficient: $\frac 1 5 y = \frac {1y}{5}$, $\frac 3 7 x = \frac {3x}{7}$.

Like Terms

Terms with the same variables and exponents are alike and should be combined. The terms of $4y^3 - 3xy + x -9$ are:
$4y^3$
$-3xy$
$x$
$-9$

Terms containing varialbes are variable terms. Those without variables are constant terms. Variable terms are made up of the variable part and the coefficient. The coefficient is the multiplicative factor in front of the variable. $x$ implies $1 \times x$ and $-x$ implies $-1 \times x$. The expression $13ab + 4 - 3ab - 10$ has the following sets of like terms:
${ 13ab, -3ab }$
${ +4, -10 }$

All constant terms are like terms.
The terms $3y$ and $2y^2$ are not like terms!

Like terms are combined by adding their coefficients: $13ab + (-3ab) = 10ab$. A term that simplifies to $0$ can be omitted.

Simplifications for multiplications:
$5(4x) = (5 \times 4) \times x = 20x$
$(-3)(-4y) = 12y$

Distribution

If the inside of parantheses is an expression, the multiplication's distributive property helps in simplifying:
$2(4 + x) = 2 \times 4 + 2 \times x = 8 + 2x = 10x$
$-4(2a + 3) = -4(2a) + (-4)3 = -8a + (-12) = -8a -12$ $-(x+y) = -1 \times x + -1 \times y = -x + -y$

In complex expressions distribution comes before simplification of like terms. In equations, like terms can only be combined if they are on the same side of the equation:
$x - 7 = -2 - 6 \Harr x - 7 = -8$

$(y^2 + 3) (y + 5)$ | this is also solved by making use of the distributive property.

$(y^2 * y) + (y^2 + 5) + (3y) + (15)$

$y^3 + y^2 + 5 + 3y + 15$

$y^3 + y^2 + 20 + 3y$

Eliminating Decimals

Equations containing decimals can be cleaned up by multiplying both sides by the place value furthest to the right in any number of the equation:

$-9.x - 0.05 = 10.5x + 1.05$ | $\times 100$
$-950x - 5 = 1050x + 105$

Because the furthest place value of 0.05 and 1.05 is the hundreth.