Algebra II - Simplifying Equations

Simplify.

Explanation

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:

Betty is working on increasing the number of minutes that she can run consecutively. Each time she goes for a jog she runs minutes longer than the previous time. She starts at minutes per jog. If she runs times per week, how long can she run after weeks?

$32 \text{ minutes}$

$24 \text{ minutes}$

$34 \text{ minutes}$

$40 \text{ minutes}$

$38 \text{ minutes}$

Explanation

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:

Solve for :

$x= \frac{2}{2y+2z+1}$

$x= \frac{2}{2y+2z-1}$

$x=\frac{2-y-z}{3}$

$x=\frac{2}{3y+3z}$

$x=\frac{2-x}{2y+2z}$

Explanation

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: $x= \frac{2}{2y+2z+1}$

Brian places an order for pairs of socks and belts at a total cost of . Danny places an order at the same store for pairs of socks and belts, at a total of . What is the cost for a single pair of socks and a single belt?

$\text{socks} = $5; \text{ belt} = $10$

$\text{socks} = $2.75; \text{ belt} = $16$

$\text{socks} = $9.54; \text{ belt} = $4.55$

$\text{socks} = $10; \text{ belt} = $5$

$\text{socks} = $6.70; \text{ belt} = $5.50$

Explanation

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.

Simplify the equation and solve:

$\frac{16}{3}$

$\frac{3}{4}$

Explanation

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: $x=\frac{16}{3}$

They say it takes , hours of practice to become an expert at something. If Anna practices piano times a week for minutes and Maggie practices piano times a week for an hour, who will hit , hours first? How many weeks will it take each girl?

Anna; Anna will take weeks and Maggie will take weeks.

Maggie; Maggie will take weeks and Anna will take weeks.

Anna; Anna will take weeks and Maggie will take weeks.

Maggie; Maggie will take weeks and Anna will take weeks.

Anna; Anna will take weeks and Maggie will take weeks.

Explanation

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

$45 \times 6=270 \text{ min}$

$\frac{270 \text{ min}}{60}=4.5 \text{ hours}$

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:

$\frac{10,000}{4.5}\approx 2,222$

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:

$30\times 7=210 \text{ min}$

$\frac{210 \text{ min}}{60}=3.5 \text{ hours}$

$\frac{10,000}{3.5}\approx 2,857$

Simplify

Explanation

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: .

Simplify the equation:

$\frac{17}{3}$

$-\frac{17}{3}$

$\frac{1}{7}$

Explanation

Add 38 on both sides.

Divide by two on both sides.

$\frac{2(3x-2) }{2}= \frac{38}{2}$

The equation becomes:

Add two on both sides.

Divide by three on both sides.

$\frac{3x}{3}=\frac{21}{3}$

The answer is:

Simplify:

$\frac{25x^{2}y^{3}}{20x^{2}y^{6}}$

$\frac{5}{4y^{3}}$

$\frac{4y^{3}}{5}$

$\frac{5y^{3}}{4}$

$\frac{5y^{3}}{4y^{6}}$

$\frac{5}{4y^{2}}$

Explanation

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:

$\frac{5*5*x*x*y*y*y}{4*5*x*x*y*y*y*y*y*y}$

After reducing, we will get

$\frac{5}{4*y*y*y} = \frac{5}{4y^{3}}$

Dave starts biking at miles per hour. an hour later, his friend Mike starts biking after him from the same starting point at miles per hour. How long does it take Mike to catch up to Dave?

$5/6 \text{ hour}$

$2 \text{ hours}$

Mike will never catch up to Dave.

$0.3125 \text{ hour}$

$1 \text{ hour}$

Explanation

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: