Simplifying Equations - Algebra II

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Question

Simplify.

Answer

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:

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Question

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?

Answer

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:

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Question

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?

Answer

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:

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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.

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Question

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?

Answer

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:

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

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

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

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Question

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?

Answer

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:

$t=\frac{4}{3}\text{ hours}$

"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.

$\frac{4}{3}-\frac{1}{2}=\frac{8}{6}-\frac{3}{6}=\frac{5}{6}$

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

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Question

Simplify

Answer

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: .

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Question

Simplify and solve:

Answer

Add on both sides of the equation.

Subtract four from both sides.

Divide by twelve on both sides.

$\frac{2}{12}=\frac{12x}{12}$

Reduce.

$x=\frac{1}{6}$

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Question

Simplify the equation and solve:

Answer

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

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Question

Solve for :

Answer

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

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Question

Simplify:

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

Answer

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:

$\frac{55xxyyy}{45xxyyyyyy}$

After reducing, we will get

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

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Question

Simplify:

$\frac{4x^{2}}{2x}$

Answer

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:

$\frac{22xx}{2x}$

After reducing, we will get

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Question

Simplify:

$\frac{2x^{2}+24x}{2x}$

Answer

To simplify:

$\frac{2x^{2}+24x}{2x}$

1. Factor out from $2x^{2}+24x$ and we will get:

$\frac{2x(x+12)}{2x}$

2. Reduce the equation and the final solution is:

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Question

Simplify the expression by combining like terms.

Simplify:

Answer

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.

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Question

Solve for : $\frac{x(x^3+3y)}{x+2} = 2x$

Answer

Multiply on both sides in order to eliminate the denominator.

$\frac{x(x^3+3y)}{x+2} \cdot (x+2)= 2x\cdot (x+2)$

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.

$\frac{3xy}{3x} = \frac{-x^4+2x^2+4x}{3x}$

Simplify both sides.

The answer is: $y=-\frac{1}{3}x^3+\frac{2}{3}x+\frac{4}{3}$

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Question

Simplify the equation:

Answer

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

$\frac{6(2x-3)}{6}=\frac{-48}{6}$

Simplify both sides.

Add three on both sides.

Simplify both sides.

Divide both sides by two.

$\frac{2x}{2}=\frac{-5}{2}$

The answer is: $x=-\frac{5}{2}$

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Question

Simplify this expression:

$(4x^{4}-x^{2})+(x^2+5x^4)$

Answer

Drop the parentheses and combine like terms.

$5x^{4}+4x^{4}=9x^{4}$

and

$-x^{2}+x^{2}=0$

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Question

Simplify the following equation:

Answer

Subtract nine from both sides.

Divide both sides.

$\frac{3x}{3}=\frac{-66}{3}$

The answer is:

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Question

Simplify the equation:

Answer

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:

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Question

Simplify the equation: $(\frac{2}{5}x)^2+\frac{1}{2}x-\frac{2}{5}x-8$

Answer

$(\frac{2}{5}x)^2+\frac{1}{2}x-\frac{2}{5}x-8$

Start by squaring everything within the parentheses:

$\frac{4}{25}x^2+\frac{1}{2}x-\frac{2}{5}x-8$

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

$\frac{4}{25}x^2+\frac{5}{5}\frac{1}{2}x-\frac{2}{2}\frac{2}{5}x-8$

$\frac{4}{25}x^2+\frac{5}{10}x-\frac{4}{10}x-8\rightarrow \boldsymbol{\frac{4}{25}x^2+\frac{1}{10}x-8}$

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