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# Linear Equations

## Isolate multi-variable expressions for a given variable

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Simplifying and Solving for Variables in Algebraic Equations

### Simplifying Expressions

TIP: To simplify algebraic expressions, use the technique of "bucketing".  By "bucketing," we mean gathering all the X's together in one bucket and putting all the Y's in another bucket. This allows you to reduce the number of terms in the expression, as often variables and numbers are repeated. The more formal name for bucketing is combining like terms.To combine like terms, simply add or subtract where necessary.

#### Guided Practice

Let's practice combining like terms.

Simplify \begin{align*}w^2+9-4w^2+3w^4-7w-11\end{align*} .

1. Combine whole numbers first.
1. \begin{align*}9-11=-2\end{align*}
2. Cross out these whole numbers in the original expression to remember that you have already accounted for them. Although this step is not listed again below, you should repeat this step after every time you combine like terms.
1. 9-........11
3. Combine the \begin{align*}\sim w^2\end{align*} terms.
1. \begin{align*}w^2-4w^2=-3w^2\end{align*}
1. Note that you cannot combine the ~\begin{align*}w^2\end{align*} terms with the ~\begin{align*}w^4\end{align*} terms because they are not like terms.
4. Combine the \begin{align*}w\end{align*}terms.
1. (In this case we can't.) \begin{align*}-7w=-7w\end{align*}
5. Combine \begin{align*}w^4\end{align*} terms.
1. (In this case we can't.) \begin{align*}3w^4=3w^4\end{align*}
6. Check that you have accounted for all the terms in your expression.  In other words, make sure all the terms in your expression are crossed out.
7. In the turquoise expressionsabove, add up all of the terms to the right of the equals sign.
1. \begin{align*}3w^4-3w^2-7w-2\end{align*}
8. Make sure you have listed all the powers from greatest to least.
1. \begin{align*}3w^4-3w^2-7w-2\end{align*}
9. If you can pull out a greatest common factor out of all of the terms, do so.

#### Vocabulary

 __________ A number that is added or subtracted within an expression. Greatest Common Factor ______________________________________________

#### Practice

Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”

1. \begin{align*}5b-15b+8d+7d\end{align*}
2. \begin{align*}6-11c+5c-18\end{align*}
3. \begin{align*}3g^2-7g^2+9+12\end{align*}
4. \begin{align*}8u^2+5u-3u^2-9u+14\end{align*}
5. \begin{align*}2a-5f\end{align*}
6. \begin{align*}7p-p^2+9p+q^2-16-5q^2+6\end{align*}
7. \begin{align*}20x-6-13x+19\end{align*}
8. \begin{align*}8n-2-5n^2+9n+14\end{align*}

Factor out the greatest common factor out of the following.

1. \begin{align*}6a-18\end{align*}
2. \begin{align*}9x^2-15\end{align*}
3. \begin{align*}14d+7\end{align*}
4. \begin{align*}3x-24y+21\end{align*}

Challenge: We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.

1. \begin{align*}2b^2-5b\end{align*}
2. \begin{align*}m^3-6m^2+11m\end{align*}
3. \begin{align*}4y^4-12y^3-8y^2\end{align*}

### Solving Algebraic Equations for a Variable

TIP: The key to solving algebraic equations for a variable is isolating that variable. If there is only one variable (i.e x) in your equation, this is simple to do.  If there are multiple variables (i.e x&y), you will have to use systems of equations (described in another concept) or solve for your variable in terms of another.

Isolating a variable requires that you "undo" the operations in an equation.  For example, if there is a "-12x" on the right side of the equals sign and you want to rid this term from this side, you need to perform the opposite operation on both sides of the equation. In this case, this would be adding 12x to both sides of the equation.  In algebra, you can never perform an operation on only one side of the equation; it always applies to both.

#### Practice

In some of these cases, whatever you solve for may be in terms of another variable. (i.e \begin{align*}x=\frac{3}{4}y\end{align*})

Solve the following equations or formulas for the indicated variable.

1. \begin{align*}6x-3y=9\end{align*} ; solve for \begin{align*}y\end{align*} .
2. \begin{align*}4c+9d=16\end{align*} ; solve for \begin{align*}c\end{align*} .
3. \begin{align*}5f-6g=14\end{align*} ; solve for \begin{align*}f\end{align*} .
4. \begin{align*}\frac{1}{3}x+5y=1\end{align*} ; solve for \begin{align*}x\end{align*} .
5. \begin{align*}\frac{4}{5}m+\frac{2}{3}n=24\end{align*} ; solve for \begin{align*}m\end{align*} .
6. \begin{align*}\frac{4}{5}m+\frac{2}{3}n=24\end{align*} ; solve for \begin{align*}n\end{align*} .
7. \begin{align*}P=2l+2w\end{align*} ; solve for \begin{align*}w\end{align*} .
8. \begin{align*}F=\frac{9}{5}C+32\end{align*} ; solve for \begin{align*}C\end{align*} .

Find the value of \begin{align*}y\end{align*} given the value of \begin{align*}x\end{align*} .

1. \begin{align*}4x-8y=2; x = -1\end{align*}
2. \begin{align*}2y-5x=12; x = 16\end{align*}
3. \begin{align*}3x+\frac{1}{2}y=-5; x = 7\end{align*}
4. \begin{align*}\frac{1}{4}x+\frac{2}{3}y-18=0; x = -24\end{align*}
5. 

1. Solve the equation for \begin{align*}h\end{align*} .
2. Find \begin{align*}h\end{align*} if the surface area is \begin{align*}120\pi \ cm^2\end{align*} and the radius is 6 cm.
3. Challenge The formula for the volume of a sphere is \begin{align*}V=\frac{4}{3}\pi r^3\end{align*} . Solve the equation for \begin{align*}r\end{align*} , the radius.



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