Solving Equations

SOLVING SIMPLE EQUATIONS

 

 

To solve with opposite operations I would perform an operation to both sides of an equation. For example, if one side has an additional 4, I would subtract 4 from both sides. Since the operations +4 and -4 cancel out, the +4 on the left side becomes +0.

 

[‘x+4=7’]

[‘x+4-4=7-4’]

[‘x+0=7-4’]

[‘x=3’]

 

Likewise, if the equations has -4 on one side, I would add +4 to both sides.

 

[‘x-4=3’]

[‘x-4+4=3+4’]

[‘x=7’]

 

The opposite operation of multiplication is division. If I want to remove multiplication from one side, I would divide both sides by the magnitude of the multiplication. You can cancel multiplication with division and vice versa.

Remember, you must always multiply or divide the entire value on both sides of the brackets.

 

[‘4x=12’]

[‘\frac4/4=\frac124’]

[‘x=3’]

 

Simmilarily, you can solve division with multiplication.

 

[‘[‘\frac4=3’]

[‘\frac4=3(4)’]

[‘x=12’]

 

Once you are proficient you will no longer need to perform the inverse on both sides, and will instead perform the inverse to only the other side. There is no functional difference in doing this, it is only to conserve space and for aesthetic reasons. The middle step is inferred

 

[‘[‘m+8=11’]

[‘m=11-8’]

[‘m=3’]

 

SOLVING MULTI-STEP EQUATIONS

 

If I have to solve a two-step equation, I must perform the inverse of the order of operations to solve.

 

For example, I must first perform addition and subtraction, and then perform multiplication and division.

 

[‘[‘6r+3=33’]

[‘6r=30’]

[‘r=5’]

 

Some equations require me to collect like terms, this is done the same was as with polynomials.

[‘3x+2=2x-4’]

[‘3x-2x=(-4)-2’]

[‘1x=(-6)’]

 

Sometimes, the distributive property must be applied. Equations like this are solved the same was as with polynomials.

 

[‘5(y+2)=20’]

[‘5y+10=20’]

[‘5y=10’]

[‘y=2’]

 

Some word problems require me to have multiple variables. Most of the time, these would be, for example, x, x+1, x+2, and x+3. Remember to use the smallest value as the base, and then perform operations based on the smallest value.

 

A triangle has angle measures that are related as follows:

 

Let x represent the smallest angle. The other angles are double and triple is value: 2x and 3x

[‘x+2x+3x=180’]

[‘6x=180’]

[‘6x/6=180/6’]

[‘x=30’]

 

Therefore, the three angles are 30, 60, and 90.

I can always check that I have the right answer by replacing the variable with the value of that variable at any step.

 

[‘2x+7=21’]

[‘2x=21-7’]

[‘2x=14’]

[‘x=\frac142’]

[‘x=7’]

 

[‘2(7)+7=21’]

[‘14+7=21’]

[‘21=21’]