How to find the solution to an equation - Algebra 1

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Question

What is ?

Answer

The key to solving this question is noticing that we can factor out a 2:

2_x_ + 6_y_ = 44 is the same as 2(x + 3_y_) = 44.

Therefore, x + 3_y_ = 22.

In this case, x + 3_y_ + 33 is the same as 22 + 33, or 55.

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Question

Simplify the result of the following steps, to be completed in order:

1. Add to

2. Multiply the sum by

3. Add to the product

4. Subtract from the sum

Answer

Step 1: 7_x_ + 3_y_

Step 2: 4 * (7_x_ + 3_y_) = 28_x_ + 12_y_

Step 3: 28_x_ + 12_y_ + x = 29_x_ + 12_y_

Step 4: 29_x_ + 12_y_ – (x – y) = 29_x_ + 12_y_ – x + y = 28_x_ + 13_y_

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Question

Solve for .

Answer

Subtract x from both sides of the second equation.

Divide both sides by to get $y=\frac{18-x}{5}$ .

Plug in y to the other equation.

$5x+15= 10*\frac{18-x}{5}$

Divide 10 by 5 to eliminate the fraction, yielding .

Distribute the 2 to get .

Add to each side, and subtract 15 from each side to get .

Divide both sides by 7 to get $\frac{x}{7}=\frac{21}{7}$ , which simplifies to .

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Question

Solve for , given the equation below.

$\frac{x + 2}{x - 1} = \frac{2x + 4}{x}$

Answer

$\frac{x + 2}{x - 1} = \frac{2x + 4}{x}$

Begin by cross-multiplying.

Distribute the on the left side and expand the polynomial on the right.

Combine like terms and rearrange to set the equation equal to zero.

Now we can isolate and solve for by adding to both sides.

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Question

Solve for :.

Answer

First factor the expression by pulling out :

Factor the expression in parentheses by recognizing that it is a difference of squares:

Set each term equal to 0 and solve for the x values:

$2x=0 \rightarrow x=0$

$3x+2=0\rightarrow x=-\frac{2}{3}$

$3x-2=0\rightarrow x=\frac{2}{3}$

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Question

Solve for :

Answer

can be simplified to become

Then, you can further simplify by adding 5 and to both sides to get .

Then, you can divide both sides by 5 to get .

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Question

Solve for :

$(x-3)^{2}=36$

Answer

To solve this equation, you must first eliminate the exponent from the by taking the square root of both sides:

$\sqrt{(x-3)^{2}}=\sqrt{36}$

Since the square root of 36 could be either or , there must be 2 values of . So, solve for

and

to get solutions of .

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Question

Solve the following equation for :

Answer

The first step is to distribute (multiply) the 2 through the parentheses:

Then isolate on the left side of the equation. Subtract the 10 from the left and right side.

Finally, to isolate , divide the left side by 2 so that the 2 cancels out. Then divide by 2 on the right side as well.

You can verify this answer by plugging the into the original equation.

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Question

Solve for :

Answer

To solve for , you must isolate it from the other variables. Start by adding to both sides to give you . Now, you need only to divide from both sides to completely isolate . This gives you a solution of $x=\frac{cd+b}{a}$ .

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Question

Solve for :

Answer

First, you must multiply the left side of the equation using the distributive property.

This gives you .

Next, subtract from both sides to get .

Then, divide both sides by to get .

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Question

Solve for :

Answer

Combine like terms on the left side of the equation:

Use the distributive property to simplify the right side of the equation:

Next, move the 's to one side and the integers to the other side:

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Question

Solve for :

Answer

To solve for , you must first combine the 's on the right side of the equation. This will give you $\ 6x-1=9x+8$ .

Then, subtract and from both sides of the equation to get .

Finally, divide both sides by to get the solution .

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Question

Solve for :

Answer

To solve for , you must isolate it so that all of the other variables are on the other side of the equation. To do this, first subtract from both sides to get . Then, divide both sides by to get $x=\frac{g-f}{e}$ .

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Question

$\frac{4^2+6\cdot 9}{4+3(2)}+\frac{7}{14}\cdot\frac{50}{25}-1=$

Answer

1. First simplify the first expression:

$\frac{4^2 +6\cdot 9}{4+3(2)}=\frac{4\cdot 4+54}{4+6}=\frac{16+54}{10}=\frac{70}{10}=7$

2. Then, simplify the next two expressions:

$\frac{7}{14}\cdot\frac{50}{25}=\frac{1}{2}\cdot\frac{2}{1}=1$

3. Finally, add and subtract:

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Question

Answer

1. First, simplify the numerator by finding a common denominator:

$\frac{6}{3}-\frac{4}{12}=\frac{6}{3}-\frac{1}{3}=\frac{5}{3}$

2. Next, simplify the denominator. These fractions can be added together without any additional work:

$\frac{9}{5}+\frac{6}{5}= \frac{15}{5}=3$

3. Then, simplify by multiplying top and bottom by the recriprocal of the denominator:

$\frac{\frac{5}{3}}{3}\cdot\frac{\frac{1}{3}}{\frac{1}{3}}=\frac{\frac{5}{9}}{1}=\frac{5}{9}$

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Question

Solve for x.

$\frac{x^2-x^3-(-x-4)}{\frac{1}{x}+\frac{x}{1}}, x=-2$

Answer

1. First solve for the numerator by plugging in -2 for x:

2. Then, solve the denominator by combining the fractions:

$\frac{1}{x}+\frac{x}{1}= \frac{-1}{2}+\frac{-2}{1}=\frac{-1}{2}+\frac{-4}{2}=\frac{-5}{2}$

3. Finally, "rationalize" the complex fraction by multiplying top and bottom by -2/5:

$\frac{14}{\frac{-5}{2}}\cdot\frac{\frac{-2}{5}}{\frac{-2}{5}}=\frac{\frac{-28}{5}}{1}=\frac{-28}{5}$

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Question

$A\ \blacksquare\ B= \frac{A + B}{2}-\frac{2}{A-B}$

If $8\ \blacksquare\ B = 6$ , find the value of .

Answer

The simplest way to solve this is to test the answer choices provided by plugging them in the equation. Start with the number in the middle, which in this case is 6. Replace every "B" in the equation with "6", and solve.

$\frac{8+(6)}{2}-\frac{2}{8-(6)}=\frac{14}{2}-\frac{2}{2}=7-1=6$

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Question

$A\bigtriangledown B = \frac{-A^2+3(-B)^2}{\frac{2}{A}+\frac{1}B{}}$

$4\bigtriangledown 2=$

Answer

Here, . Plug these values in and simplify.

Start with the numerator first:

Then do the same in the denominator:

$\frac{2}{A}+\frac{1}{B}= \frac{2}{4}+\frac{1}{2}= \frac{1}{2}+\frac{1}{2}=1$

Finally, combine the two results to find the solution:

$\frac{-4}{1}=-4$

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Question

What number is six less than eight more than one half of four times the square of the greatest negative integer?

Answer

Turn the word problem into math. Start at the end--what is the greatest negative integer? -1!

$(-1)^2\cdot4\cdot\frac{1}{2}+8-6=1\cdot4\cdot\frac{1}{2}+8-6=2+8-6=4$

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Question

$18\div\frac{1}{6}\cdot\frac{9}{72}\cdot\frac{4}{27}\cdot(-8+7)^9=$

Answer

$18\div\frac{1}{6}\cdot\frac{9}{72}\cdot\frac{4}{27}\cdot(-8+7)^9=18\cdot6\cdot\frac{1}{8}\cdot\frac{4}{27}\cdot-1$

Remember to cancel as you go--the 18 will cancel with the 27, the 4 will cancel with the 8, and so on:

$18\cdot6\cdot\frac{1}{8}\cdot\frac{4}{27}\cdot-1= 2\cdot6\cdot\frac{1}{2}\cdot\frac{1}{3}\cdot-1$

Continue to reduce:

$2\cdot6\cdot\frac{1}{2}\cdot\frac{1}{3}\cdot-1= 2\cdot1\cdot-1=-2$

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