Help with Very Basic Balancing of Equations

[kolacka]

Hi!

I'm learning pre-algebra before I get stuck into algebra.

So far I understand that what you do to one side of an equation you must do to the other side.

I'm struggling to understand why the calculations then are different for this problem, solved 2 ways:

Equation: 7x -2 = -10

Solution 1: 7x/7 - 2 = -10/7

x = -10/7 + 2

x = .571

Solution 2: 7x +2 = -10 + 2

7x = -8

7x/7 = -8/7

x = -1.142


I'm confused because I do the same to both sides in either solution - am I doing something wrong? Does it matter whether I plus / minus or divide/multiply first? Why is the answer different if I divide first, and then take the -2 to the other side?

Thank you :/

 

[JeffM]

Some vocabulary and concepts.

An equation consists of two expressions joined by an equal sign. A valid equation states truly that each expression represents the SAME number.

The idea behind balancing equations is that if we change one expression to a new number, say by dividing it by 7, we can maintain equality by making the same change to the other expression because both expressions represent the same number. If you divide one expression by 7, you can maintain equality only by dividing the other expression by 7.

\(\displaystyle 7x + 2 = -\ 10 \implies \dfrac{7x + 2}{7} = -\ \dfrac{10}{7} \implies\)

\(\displaystyle \dfrac{7x}{7} + \dfrac{2}{7} = -\ \dfrac{10}{7} \implies x + \dfrac{2}{7} - \dfrac{2}{7} = -\ \dfrac{10}{7} - \dfrac{2}{7}\implies\)

\(\displaystyle x = -\ \dfrac{12}{7}.\)

Although it is not logically necessary, you will avoid mistakes by gathering all terms containing the unknown on one side of the equation and all terms not including the unknown on the other side of the equation before doing any division. This is called "isolating the unknown."

\(\displaystyle 7x + 2 = -\ 10 \implies 7x + 2 - 2 = -\ 10 - 2 \implies\)

\(\displaystyle 7x = -\ 12 \implies \dfrac{7x}{7} = -\ \dfrac{12}{7} \implies x = -\ \dfrac{12}{7}.\)

Same result either way, but isolating the unknown is simpler and so less prone to error.

 

[mmm4444bot]

Does it matter whether I plus / minus or divide/multiply first?

As JeffM noted, it does not matter, but we normally do the additions/subtractions of terms first, and finish with the division.

Solution 1: 7x/7 - 2 = -10/7

As y2kevin noted, you only divided part of the left-hand side by 7. When dividing each side of an equation, the entire expression (on each side) needs to be divided.

x = -10/7 + 2

x = .571

Do not switch to decimal form; work with exact expressions. When you round decimal numbers, your solutions will not be exact.

Knowing how to add fractions and whole numbers together is a pre-algebra topic. Make sure you know how to do arithmetic with fractions, before starting algebra.

 

[kolacka]

Thank-you all! I will take on your advice and will spend time now re-vising my methods!

I was very bad at maths in high school due to a total lack of confidence, then conquered my fear in my final year of university and did differential calculus, quadratic equations, etc, for my economics degree. However, I haven't done any maths for 6 years and I'm back to re-learning the basics!

I'm excited, but also quite flustered

I appreciate all of your replies and will save them for reference. You'll be hearing more from me