\(\displaystyle 5h + 7k = 12\)

\(\displaystyle 5h + 7k - 7k = 12- 7k\)

\(\displaystyle 5h = 12- 7k\)

\(\displaystyle \frac{5h}{5} =\frac{ 12- 7k}{5}\)

\(\displaystyle h =\frac{ - 7k+12}{5}\)

\(\displaystyle 2x+8=13\)

Solve equation by using inverse order of operations to get \(\displaystyle x\) by itself.

Order of operations: PEMDAS. Addition and subtraction usually come last, so when using inverse order of operations to "undo an expression," they come first. 8 is being added to \(\displaystyle x\), so subtract 8 from both sides in order to get \(\displaystyle x\) alone.

\(\displaystyle 2x+8=13\)

      \(\displaystyle -8\)

\(\displaystyle 2x=5\)

Next comes multiplication/division. \(\displaystyle x\) is being multiplied by 2, so divide both sides by 2 to get \(\displaystyle x\) on its own.

\(\displaystyle \frac{2x}{2}=\frac{5}{2}\)

\(\displaystyle x=\frac{5}{2}\)

\(\displaystyle -3x+20=8\)

1) Get \(\displaystyle x\) by itself. The first step, then, is to use order of operations backwards to decide where to start.

Take PEMDAS and flip it around. So subtraction/addition comes first, and therefore the 20 has to be the first to go.

2) Move the 20 to the other side of the equation using inverse operations. 20 is added to the \(\displaystyle x\) in the problem, so it needs to be subtracted from both sides.

\(\displaystyle -3x=8-20\)

\(\displaystyle -3x=-12\)

3) Divide by \(\displaystyle -3\) to get the \(\displaystyle x\) alone.

\(\displaystyle x=4\)

\(\displaystyle -5x-13=-8\)

1) The first step in solving an equation is always to simplify as much as possible. Is there a factor that all terms have in common? They all have a negative sign, so \(\displaystyle -1\) can be factored out. It is not necessary to remove the \(\displaystyle -1\); however, the problem will be much simpler to solve without the negative signs to add confusion.

So divide both sides by \(\displaystyle -1\):

\(\displaystyle 5x+13=8\)

2) Next, get \(\displaystyle x\) by itself using inverse order of operations. So subtract 13 from both sides and combine like terms:

\(\displaystyle 5x=8-13\)

\(\displaystyle 5x=-5\)

Then divide both sides by five:

\(\displaystyle x=-1\)

1) First, define your variables:

\(\displaystyle x=\) number of hours the truck is in use

\(\displaystyle P=\) total price

Even though the question gives the definitions, it's always helpful to rewrite them.

2) A flat price is the part of the price that is dependent on nothing. So the flat price stands alone:

\(\displaystyle P=175+. . .\)

3) The amount of money that the $20 rate will demand from the renter is dependent on the number of hours the truck is in use, as it is a rate and therefore involves some sort of time. Thus it is clear from looking at the definition of our variables that the 20 must be multiplied by \(\displaystyle x\) to determine how much more the renter must pay for the truck's use, on top of the $175. The two numbers must be added together to find the final price, which includes both.

\(\displaystyle P=175+20x\)

First, use the distributive property to simplify the right side of the equation: 

\(\displaystyle 3x-3=6x+12\).

Next, subtract \(\displaystyle 6x\) and add \(\displaystyle 3\) to both sides to get 

\(\displaystyle -3x=15\).

Finally, divide both sides by \(\displaystyle -3\) to get 

\(\displaystyle x=-5\)

Use the distributive property to get \(\displaystyle 3x-15=21\). Isolate for \(\displaystyle x\) and you get 12. 

\(\displaystyle 19=\frac{4}{w}+3\)

Subtract 3 from both sides of the equation.

\(\displaystyle 16=\frac{4}{w}\)

Take the reciprocal of both sides of the equation

\(\displaystyle \frac{1}{16}=\frac{w}{4}\)

Multiply both sides by 4

\(\displaystyle w=\frac{4}{16}=\frac{1}{4}\)

Add 33 to both sides of equation

\(\displaystyle x^{2}=25\)

Take the square root of both sides of equation

\(\displaystyle \sqrt{x^{2}}=\sqrt{25}\)

\(\displaystyle x=5\)

A yard stick has 36 inches.  Let \(\displaystyle x=\) length of piece one.  

Then, let \(\displaystyle 2x+6=\) length of piece two.

The length of both pieces has to be 36 inches.

So, \(\displaystyle x+(2x+6)=36\)

After combining like terms we get: \(\displaystyle 3x+6=36\)

Subtracting 6 from both sides gives: \(\displaystyle 3x=30\)

Dividing both sides by 3 gives:\(\displaystyle x=10\), the length of piece one.

The length of piece two is given by \(\displaystyle 2x+6\) 

Substituting in \(\displaystyle x=10\) gives: \(\displaystyle 2(10)+6=26\)

Therefore, the lengths of the two pieces are 10 and 26 inches.