Solving Linear Equations |

**LINEAR EQUATIONS **

**This is an Equation**

Now that we've learned the syntax, order and grammar rules of the language known as algebra, we can use it to write **sentences or equations** to describe a myriad of situations. Equations are used in Physics, Chemistry, Biology, Computer Programing, Banking, --- you name it. Why, I used a simple algebraic equation one morning to calculate the mix of water and 10% cream I needed to supply enough whole milk (3.5%) for my bowl of breakfast cereal. The timing of traffic light changes is determined by equations based on statistical analysis of traffic flow.

An algebraic equation works exactly the same way balanced scales do, with the equal sign replacing the pivot point of the scales. Whatever you do to one side of the equation must be done to the other side in order to **maintain equality or balance**. Those are the operational words. We must **maintain equality at all times** -- why do you think we call it an **EQUATION**?? We would call it something else if it was something else -- but it's not. There are other types of statements in algebra such as inequalities -- (coming up in lesson 7) -- and functions -- which are equations but different and the same. In this lesson, we concentrate on the **equality relation.**

A binary relation is a relation between two terms or expressions. For example, *a* = 3 is a binary relation statement. The relation symbol is = (equality) and the ** "binary"** part -- binary meaning 2 things are involved -- are the

There are **three binary relations in mathematics**.

They are: ..... **equal** to ( = ), ..... **greater** than ( > )..... and ..... **less** than ( < ).

We begin with equations: **statements of equality**. Then we'll investigate the **less than** and **greater than** relationships -- the inequalities in subsequent algebra lessons.

A **linear equation** is one in which all the **variables** are **to the first power**. Should we decide to graph a **linear **equation, we would get a **line**. To solve a linear equation, we simply **isolate the variable** for which we wish to solve, and perform the correct algebraic operations to get a statement about its value.

**Whatever we do to one side of the equation, we must do to the other.**

**Example:** Solve ** x + 7 = 10** . To solve, we must

Our last statement in the solution will be

If we could take away the 7 from the left side, we'd be left with what we want, so that's what we do. Since it's an equation and we must always maintain equality, we have to

Since

Notice how the "7" seemed to jump over the equal sign and in doing so, changed from positive to negative.

Had our equation been * x* – 7 = 10**,** we would **add 7 to both sides**. We would get:

*x* – 7 __+ 7__** = 10 **** + 7** which becomes

and when we check, we get

Again, the "**– 7**" seemed to jump over the equal sign, but this time, it changed from negative to positive. To shorten the solving process, we use a technique called **transposing**.

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**Transposing: shortcut to solving equations**

We often refer to adding and subtracting terms in an equation as transposing.

We use it to move algebraic terms from one side of the equal sign to the other -- by changing their sign.

Here's an example.

Solve: **5 x **

We'll move the **3 x** over to the left side, and we'll move the

**5 x **

we collect like terms to get:.....

Notice how **3 x**

Here's a more complicated example. First we'll simplify the expressions in the equation, then we'll collect like terms and only then will we transpose terms to organize the setup for the solution.

Solve for *x* if **3(3***x*** – 1) = 4(***x*** – 3) – 1**

Step 1: remove brackets ..... ..... ..... ..... ** 9***x*** – 3 = 4***x*** – 12 – 1**

Step 2: collect terms (– 12 and – 1) ..... .. **9***x*** ****– 3**** = ****4***x*** – 13**

Step 3: **transpose** **– 3** and **4 x**. ... ..... .....

Step 4: collect terms ..... ..... ..... ..... ... **5 x**

When **transposing**, we must **change the sign of any term that crosses the equal sign.**

Notice that the **– 3**** **became **+ 3** and the **4 x** became

Both methods give the same results since they are identical, but transposing is more efficient.

We use linear equations to solve word problems that involve variables to the first power. Some of these problems are best solved using a table or diagram in conjunction with equations. Others can be handled with just an equation preceded by a **" let "** statement, like **let x = the amount of money Joe has**. This is known as

**Practice**

**Solve these equations for all possible values of the variable:**

1) 3( *y* – 1) – 1 = 2 – 5( *y* + 5)

2) 3*x* – 4(*x* + 6) = 2[*x* – 3(5 – *x*)]

3) 3(4*x* – 1) = 9 – 4*x*

4) 5*x* – 3(*x* + 1) = 2(3*x* + 5)

5) 15*w* – 4 = 8*w* + 31

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**Solutions**

1) 3(y – 1) – 1 = 2 – 5(y + 5)3 y – 3 – 1 = 2 – 5y – 258 y = – 19. y = – 19 / 8 |
2) 3x – 4(x + 6) = 2[x – 3(5 – x)]3 x – 4x – 24 = 2[ x – 15 + 3x ]– x – 24 = 2( 4x – 15 ) = 8x – 30– 9 x = – 6. x = 2 / 3 |
3) 3(4x – 1) = 9 – 4x12 x – 3 = 9 – 4x16w = 12. w = ¾ |

4) 5x – 3(x + 1) = 2(3x + 5)5 x – 3x – 3 = 6x + 10– 4 x = 13. x = – 13/4 |
5) 15w – 4 = 8w + 317 w = 35. w = 5 |

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