Simple Equations

MHDEqns

One-Step Equations

 

Solving for an unknown pronumeral is the main subject of equations. For example,

x+3=7

The equal sign acts as a balance, which means both sides must be equally weighted, as depicted by the balance shown on the left. The idea is to find how many red balls are there in the green envelope. By inspection, one would know straight away that x=4. However, there is a more systematic principle of solving equations that could be taken to more advanced equation problems. To isolate x on one side, we take away 2 on the left side of the equal sign. However, to make the equation balanced, 2 must be taken away on the right hand side as well - shown below:

This principle is: whatever operation is applied to the right, the same must be applied to the left, or vice versa.

In addition, there is an even quicker method to solve simple equations. Using the example above, move the +3 on the left to the right, however change the sign so that it becomes -3 on the right hand side.

In the end, both yield the same results. Now, practice some more and your skills will be perfect!

 

Two-Step Equations

 

Two-step equations have similar properties like the name: two steps are needed. Example:

 

 

The final result must always be the value of a single pronumeral.

 

Equations with Pronumerals on Both Sides

 

If there are pronumerals on both sides, eliminate all the pronumerals on one side, and eliminate all the numbers on the other side. For example:

 

 

Equations with Fractions

 

There are different ways to solve this type of equation.

 

Examples:

1. Solve  \cfrac { p }{ 2 } -4=2

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One way to solve this is by changing all the denominators of every term so they are equal. Once the denominators are equal, we could treat the numerator as a normal simple equation.

\cfrac { p }{ 2 } -4=2\\ \\ \cfrac { p }{ 2 } -\cfrac { 8 }{ 2 } =\cfrac { 4 }{ 2 } \\ \\ p-8=4\\ \\ p=12

 

There are in fact many more methods to solve equations with fractions.

 

 

2. Solve  \cfrac { m }{ 4 } -\cfrac { 2-m }{ 3 } =\cfrac { 1 }{ 4 }

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Multiply all elements of the equations by LCMLowest Common Multiple to eliminate the denominators.


(solution courtesy of mathsteacher.com.au)

 

Another type of equations with fractions is where there are simple fractions on either side of the equation. One can cross multiply to solve for the solution.

 

3. Solve  \cfrac { 8 }{ 5 } \quad =\quad \cfrac { 3 }{ 2n }

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Substitution

 

Substituting values into a formula may be a part of solving an equation.

 

Examples:

 

The formula for the volume of a cylinder is given by V=\pi { r }^{ 2 }h where r is the radius and h is the height. Suppose a cylinder has a base radius of 10cm and a height of 80mm, what is its volume in { cm }^{ 3 }?

 

Remember that 80mm = 8cm

V\quad =\quad \pi { r }^{ 2 }h\\ \\ V\quad =\quad \pi \times { 10 }^{ 2 }\times 8\\ \\ V\quad =\quad 2,513{ cm }^{ 3 }