2.1: Equations with Variables on One Side
Erin, Jillian, Stephanie and Jacob went to the movies. The total bill for the tickets and snacks came to $72.00. What is an equation that represents this situation? How much should each teen pay to split the bill evenly?
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Khan Academy Slightly More Complicated Equations
Guidance
When solving any equation, your job is to find the value for the letter that makes the equation true. Solving equations with variables on one side can be done with the help of models such as a balance or algebra tiles.
When solving equations with variables on one side of the equation there is one main rule to follow: whatever you do to one side of the equals sign you must do the same to the other side of the equals sign. For example, if you add a number to the left side of an equals sign, you must add the same number to the right side of the equals sign.
Example A
\begin{align*}5a + 2 = 17\end{align*}
The problem can be solved if we think about the problem in terms of a balance. We know that the two sides are equal so the balance has to stay horizontal. We can place each side of the equation on each side of the balance.
In order to solve the equation, we have to get the variable \begin{align*}a\end{align*}
Let’s first subtract 2 from both sides to get rid of the 2 on the left.
Since 5 is multiplied by \begin{align*}a\end{align*}
If we simplify this expression, we get:
Therefore \begin{align*}a = 3\end{align*}
We can check our answer to see if we are correct by substituting our answer back into the original equation.
\begin{align*}5a + 2 &= 17\\ 5({\color{red}3}) + 2 &= 17\\ 15 + 2 &= 17\\ 17 &= 17 \ \ \end{align*}
Example B
\begin{align*}7b  7 = 42\end{align*}
Again, we can solve the problem if we think about the problem in terms of a balance (or a seesaw). We know that the two sides are equal so the balance has to stay horizontal. We can place each side of the equation on each side of the balance.
In order to solve the equation, we have to get the variable \begin{align*}b\end{align*}
Let’s first add 7 from both sides to get rid of the 7 on the left.
Since 7 is multiplied by \begin{align*}b\end{align*}
If we simplify this expression, we get:
Therefore \begin{align*}b = 7\end{align*}
We can check our answer to see if we are correct.
\begin{align*}7b  7 &= 42\\ 7({\color{red}7})  7 &= 42\\ 49  7 &= 42\\ 42 &= 42 \ \ \end{align*}
Example C
This same method can be extended by using algebra tiles. If we let rectangular tiles represent the variable, square tiles represent one unit, green tiles represent positives numbers, and white tiles represent the negative numbers, we can solve the equations using an alternate method.
The green algebra \begin{align*}x\end{align*}
To solve, add two negative tiles to the right and left hand sides. The same rule applies to this problem as to all of the previous problems. Whatever we do to one side we have to do to the other.
This leaves us with the following:
We can reorganize these to look like the following:
Organizing the remaining algebra tiles allows us to realize the answer to be \begin{align*}x = 3\end{align*}
Let’s do our check as with the previous two problems.
\begin{align*}3c + 2 &= 11\\ 3({\color{red}3}) + 2 &= 11\\ 9 + 2 &= 11\\ 11 &= 11 \ \ Y\end{align*}
Concept Problem Revisited
There are four teens going to the movies (Erin, Jillian, Stephanie, and Jacob). The total bill was $72.00. Therefore our equation is \begin{align*}4x = 72\end{align*}
\begin{align*}x &= \frac{72}{4}\\ x &= 18\end{align*}
Therefore each teen will have to pay $18.00 for their movie ticket and snack.
Vocabulary
 Constant

A constant is also a numerical coefficient but does not contain a variable. For example in the equation \begin{align*}4x + 72 = 0\end{align*}
4x+72=0 , the 72 is a constant.
 Equation
 An equation is a mathematical equation with expressions separated by an equals sign.
 Numerical Coefficient

In mathematical equations, the numerical coefficients are the numbers associated with the variable. For example, with the expression \begin{align*}4x\end{align*}
4x , 4 is the numerical coefficient and \begin{align*}x\end{align*}x is the variable.
 Variable
 A variable is an unknown quantity in a mathematical expression. It is represented by a letter. It is sometimes referred to as the literal coefficient.
Guided Practice
1. Use a model to solve for the variable in the problem \begin{align*}x5=12\end{align*}
2. Use a different model than used in question (1) to solve for the variable in the problem \begin{align*}3y+9=12\end{align*}
3. Using one of the models from the concept, solve for \begin{align*}x\end{align*}
Answers:
1. \begin{align*}x5=12\end{align*}
Therefore, \begin{align*}x=17\end{align*}
\begin{align*}x  5 &= 12\\ ({\color{red}17})  5 &= 12\\ 17  5 &= 12\\ 12 &= 12 \ \ \end{align*}
2. \begin{align*}3y+9=12\end{align*}
First you have to subtract 9 from both sides of the equation in order to start to isolate the variable.
Now, in order to get \begin{align*}y\end{align*}
Therefore, \begin{align*}y=1\end{align*}
\begin{align*}3y + 9 &= 12\\ 3({\color{red}1}) +9 &= 12\\ 3 + 9 &= 12\\ 12 &= 12 \ \ \end{align*}
3. \begin{align*}3x2x+16=3\end{align*}
You can use any method to solve this equation. Remember to isolate the \begin{align*}x\end{align*}
\begin{align*}3x2x+16 &= 3\\ x+16 &= 3\end{align*}
Now you can use any method to solve the equation. You now should just have to subtract 16 from both sides to isolate the \begin{align*}x\end{align*}
\begin{align*}x+16 {\color{red}16} &= 3 {\color{red}16}\\ x &= 19\end{align*}
Let's do a check to make sure.
\begin{align*}3x2x+16 &=3 \quad \text{(original problem)}\\ x+16 &= 3 \quad \text{(simplified problem)}\\ {\color{red}19}+16 &= 3\\ 3 &= 3 \ \ \end{align*}
Practice
Use the model of the balance to solve for each of the following variables.
 \begin{align*}a+3=5\end{align*}
 \begin{align*}2b1=5\end{align*}
 \begin{align*}4c3=9\end{align*}
 \begin{align*}2d=3\end{align*}
 \begin{align*}4=3e=2\end{align*}
Use algebra tiles to solve for each of the following variables.
 \begin{align*}x+3=14\end{align*}
 \begin{align*}2y7=5\end{align*}
 \begin{align*}3z+6=9\end{align*}
 \begin{align*}5+3x=3\end{align*}
 \begin{align*}2x+2=4\end{align*}
Use the models that you have learned to solve for the variables in the following problems.
 \begin{align*}4x+13=5\end{align*}
 \begin{align*}3x5=22\end{align*}
 \begin{align*}112x=5\end{align*}
 \begin{align*}2x4=4\end{align*}
 \begin{align*}5x+3=28\end{align*}
For each of the following models, write a problem involving an equation with a variable on one side of the equation expressed by the model and then solve for the variable.
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Description
Learning Objectives
Here you will begin your study of mathematical equations by learning how to solve equations a variable on only one side.
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Date Created:
Dec 19, 2012Last Modified:
Aug 11, 2015Vocabulary
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