Solve equations and simplify expressions
In algebra 1 we are taught that the two rules for solving equations are the addition rule and the multiplication/division rule.
The addition rule for equations tells us that the same quantity can be added to both sides of an equation without changing the solution set of the equation.
Example
$$\begin{array}{lcl} 4x-12 & = & 0\\ 4x-12+12 & = & 0+12\\ 4x & = & 12\\ \end{array}$$
Adding 12 to each side of the equation on the first line of the example is the first step in solving the equation. We did not change the solution by adding 12 to each side since both the second and third equations have the same solution. Equations that have the same solution sets are called equivalent equations.
The multiplication/division rule for equations tell us that every term on both sides of an equation can be multiplied or divided by the same term (except zero) without changing the solution set of the equation.
Example
$$\begin{array}{lcl} 4x-12 & = & 0\\ 4x-12+12 & = & 0+12\\ 4x & = & 12\\ \frac{4x}{4} & = & \frac{12}{4}\\ x & = & 3\\ \end{array}$$
When we simplify an expression we operate in the following order:
- Simplify the expressions inside parentheses, brackets, braces and fractions bars.
- Evaluate all powers.
- Do all multiplications and division from left to right.
- Do all addition and subtractions from left to right.
A useful rule is the denominator-numerator rule which states that the denominator and numerator may be multiplied by the same quantity without changing the value of the fraction.
Example
$$\frac{(2^{2}-2)}{\sqrt{2}}$$
First we simplify the expression inside the parentheses by evaluating the powers and then do the subtraction within it.
$$\frac{(4-2)}{\sqrt{2}}$$
$$\frac{(2)}{\sqrt{2}}$$
We then remove the parentheses and multiply both the denominator and the numerator by √2.
$$\frac{2\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}$$
As a last step we do all multiplications and division from left to right.
$$\frac{2\cdot \sqrt{2}}{2}$$
$$\sqrt{2}$$
Video lesson
Solve the given equation
$$12a\left(\frac{3b-b}{4a}\right)=36$$