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Linear Equations

LINEAR EQUATIONS

 

The magic of equations

Think of a number.

Add 5 to this number

Multiply it by 2

Subtract 4

Divide the result by 2

If you subtract the number you initially thought of, do you obtain 3?

 

Warm-up thought

Looking at the balanced scale, could you find out the weight of the red rectangle provided that each blue circle weighs 100 g?

Scale

Now, can you explain how did you find out the answer?

 

Practice with symbols

Draw one of the following symbols used in equations above the word which fits best:

+  -  x  /  = 

add, addition, sum, increased by, subtraction, difference, minus, decreased by, less, multiplication, product of, multiplied, times, division, quotient of, divide by, is/are, gives, equals, total.

 

Equation maker

First, numbers and letters are combined using mathematical operators. These are algebraic expressions.

An equality is a combination of two different expressions that are equivalent.

Then, an equation is an equality in which a symbol or a letter is involved.

This symbol represents an unknown value, and it is called variable.

The objective of the equation is to find the value of the variable that makes the equality be true.

 

Now, write an equation based on the following sentences. The first one is an example. The letter x is used as variable..

1. Three times a number is nine.

    3 x = 9

2. Four times a number minus seven equals nine plus minus four thimes the number.

3. Six plus a number totals five.

4. Minus twelve divided by a number is four.

5. Three plus four times a number equals minus one.

 

A break to feed the brain

Watch the following video about how to solve linear equations:

Can you now solve the equations you wrote in the equation maker exercise? If not yet, please watch the videos Solving Equations #2 and Solving Equations #3 attached to this unit.

Now, nothing stops you! Solve the equations you built previously.

 

Writing down the ideas for solving linear equations

Fill in the gaps using the words provided:

addition   Divide   equal   multiplying   check   subtraction

1. Use the distributive property to clear parentheses

2. Clear fractions by ________ each term in the equation by the lowest common divisor.

3. Use _______ and ________ to put all variable terms on one side of the equation and all constants on the other side.

4. Combine similar terms on each side of the _______ sign.

5. ________ both sides of the equation by the coefficient of x.

6. Simplify.

One final thing that you must always do is to _______ your solutions. Replace x with the solutions and check if the equality is true.

 

Practice makes the master

Find the value of x in the following linear equations. Try to identify the steps you are using.

a) 3x = 24

b) 4x + 7 = 23

c) 2 + 3x = -10

d) 2 / 4x - 1 = 3

e) 36 = 6 (x - 6)

f) 80 = 9x - 1

g) 2 (x + 4) = 12

h) 3x - 2 / 5 = 2

i) 25 - 3x = 1 + 3 (4 + 3x)

j) x / 2 + x / 5 = 10 / 7

Have you checked your answers?

 

Rule the real world

The aim of equations is to help us solve problems from the real life. Find the answer to this questions.

a) You pay 10€ each month to join a gym. Unfortunately, the cycling class is not included so you pay 3€ each time you go to this activity. At the end of the year, you have paid 171€. How many times did you go to the cycling class this year?

b) The height of three trees are three consecutive numbers and the sum of the three heights is 33. What is the height of the three trees?

c) With the money I have now, I need 1.8€ more to buy my favourite magazine. If I had two times the money I have now, I would have 2€ left after buying it. How much does the magazine cost? How much do I have? (You may use more than one equation and variable)

 

Magic unveiled

Can you now express the text in the first task The magic of equations using algebraic language and equations?

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