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Equations with variables

In this section, you will learn how to solve equations that contain unknown variables. You will learn how to solve equations mentally by using the multiplication table and you will also learn how to identify a solution to an equation with given numbers as well as by using inverse operations.

You can solve an easy equation in your head by using the multiplication table.

Example

\begin{array}{lcl} 8x=64 \end{array}

\begin{array}{lcl} 8\cdot x=64 \end{array}

Which number should you multiply 8 by to get a product of 64? By using the multiplication table, we know that the number is 8.

\\8\cdot 8=64\\

When we solve an equation, we figure out what value of x (or any other variable) makes the statement true (satisfies the equation).

Example

Which of the following numbers is a solution to the equation? x = 2, 7,  or 8?

14-x=7

Here you are given the numbers 2, 7 and 8. One of these numbers will satisfy the equation. If you don't know the solution right away, you can investigate which of the given numbers gives results in the correct answer by plugging in the different values of x.

\\\begin{matrix} x=2\Rightarrow & 14-2=12& {\color{red} Wrong}\: \: \\ x=7\Rightarrow & 14-7=7\: &{\color{green} Correct} \\ x=8\Rightarrow & 14-8=6\: & {\color{red} Wrong}\: \: \end{matrix}

So x = 7.

You have already solved equations where the solutions are quite easy to see, by using mental math or patterns. Most equations are harder to solve and you have to simplify the equation before you can see the solution. One way to do this is to use inverse operations.

An operation is, for example, addition, multiplication, division and subtraction. An inverse operation is an operation that reverses the effect of another operation. Addition and subtraction are inverses of each other, just like division and multiplication are inverses.

Example

With numbers

\begin{array}{lcl} \; \; \; \; \; 18+4=22\\ 18+4{\color{blue} \, -\, 4}=22{\color{blue} \, -\, 4}\\ \: \: \: \: \: \: \: \: \: \: \: \, \, 18=18 \end{array}

With variables and numbers

\begin{array}{lcl} \; \; \; \; \; x+4=22\\ x+4{\color{blue} \, -\, 4}=22{\color{blue} \, -\, 4}\\ \: \: \: \: \: \: \: \: \: \: \: \, \, x=18 \end{array}

We subtract 4 from both sides.

Example

With variables and numbers

\\ x\cdot 2=10\\\\ \frac{x\cdot 2}{{\color{blue} 2}}=\frac{10}{{\color{blue} 2}}\\\\ x=5 \\

We divide both sides by 2

Video lesson: Solve the following equation

\\8\cdot x-x=21

Video lesson: Solve the following equation using inverse operations

6x+4=28

Next Class:  Introducing Algebra, Coordinate system and ordered pairs