Simplifying Equations - Algebra 2

If Betty increases her time by 2 minutes 3 times a week, then she increases her time by a total of 6 minutes per week. We also know that she starts out running for 20 minutes. We can use this information to set up the equation in form as:

Here x is the number of weeks, and y is the number of minutes she can run at the end of each week. The question is asking for how many minutes she can run at the end of 2 weeks, so we plug in 2 for x:

We can set up a system of equations to solve this problem. If we call the price of a single pair of socks "s" and a single belt "b" then we can set up the following equations:

The easiest way to solve this system would be to combine them in such a way that eliminates one of the variables. We can do this by multiplying the bottom equation by -2, then adding it to the top equation.

This simplifies to:

Now when we add it to the first equation, the s variables will cancel out:

                                            

We can now solve for b, then plug that value into either one of the original equations.

Thus, we get that the cost of a belt, b, is 10 dollars and the cost of a pair of socks, s, is 5 dollars.

The number of hours that Anna practices per week can be found by:

If Anna practices for 4.5 hours per week, then the number of weeks it will take for her to reach 10,000 hours can by found by:

We can use the same process to find the number of hours that Maggie practices per week, and from there the number of weeks it will take her to reach 10,000 hours:

The equation for Dave's distance after t hours can be written as:

Since Mike is biking for half an hour less than Dave, we can write his distance as:

We want to know how many hours it takes for Mike to catch up to Dave. In other words, we are looking for the time when they have traveled the same distance. So, we can set the equations for distance traveled equal to each other:

"t" is how long Dave bikes for until Mike catches up. "t-.5" is the amount of time that Mike bikes for, and that is what we are trying to find.

So, Mike bikes for 5/6 hours until he catches up with Dave.

Rearrange the terms by grouping like terms together:

then simplify:

.

Finally, move the numerical terms to the other side of the equation so that all of the like terms are together:

, then simplify: .

Add on both sides of the equation.

Subtract four from both sides.

Divide by twelve on both sides.

Reduce.

Use order of operations to expand the terms of the parenthesis first.

Rewrite the equation.

Simplify the left side of the equation.

Add ten on both sides of the equation.

Simplify.

The answer is:

In order to isolate the variable, we will need to use order of operations to eliminate the parentheses and group the terms with an variable to one side of the equation.

Distribute inside the parentheses.

Subtract five on both sides.

Add on both sides to move the negative to the left side.

Take out a common factor of on the left side.

Divide by on both sides of the equation.

The answer is:

To simplify the equation you need to divide the numerator (top) and denominator (bottom), so:

The numerator can be factored to:

The denominator can be factored to:

After reducing, we will get

To simplify the equation you need to divide the numerator (top) and denominator (bottom), so:

The numerator can be factored to:

The denominator can be factored to:

After reducing, we will get

To simplify:

1. Factor out from and we will get:

2. Reduce the equation and the final solution is:

The original equation:

Now move all terms to one side, in this example, we will use the right side, but either side will work.

As you can see, we subtracted the terms on the left from both sides, effectively moving them to the other side.

Regrouping the terms so that "like terms" are together. Like terms are defined by having the same power of x.

Now, we add any terms that have like powers of x.

Now that all terms have been combined, we are finished. The equation is simplified.

Multiply on both sides in order to eliminate the denominator.

Simplify both sides.

Use distribution to simplify both sides.

Isolate the term with only the y-variable by subtracting on both sides.

Simplify and reorder the right side by the order of power.

Divide by on both sides.

Simplify both sides.

The answer is:

In order to simplify and solve this equation, divide by six on both sides.

Simplify both sides.

Add three on both sides.

Simplify both sides.

Divide both sides by two.

The answer is:

Drop the parentheses and combine like terms.

and

Subtract nine from both sides.

Divide both sides.

The answer is:

Add 38 on both sides.

Divide by two on both sides.

The equation becomes:

Add two on both sides.

Divide by three on both sides.

The answer is:

Start by squaring everything within the parentheses:

Now combine the fractions of degree 1 by finding a common denominator:

a. Begin by using the FOIL method to square (2x-2).

The resulting equation should be:

b. Distribute -5 to the expression inside the parentheses to get

c. Finish simplifying by combining like terms: