Solving Equations By Balancing

The two sides on a balanced scale must be equal to each other A Question of Balance The two sides on a balanced scale must be equal to each other E + 6 = 11 E = 5 What does the Egg weigh?

The left side and the right side must be balanced A Question of Balance The two sides of an equation are equal to each other The left side and the right side must be balanced 2(3) + 4 10 When you do something to one side of an equation, You have to do the same thing to the other side.

If the two sides of an equation are not equal… A Question of Balance If the two sides of an equation are not equal… 3(7) – 2 20 + 1

If the two sides of an equation are not equal… A Question of Balance If the two sides of an equation are not equal… Then it is not balanced! 3(7) – 2 20 + 1

What happens if we change one of the sides of a balanced equation? A Question of Balance What happens if we change one of the sides of a balanced equation? 8 + 3 + 1 8 + 3 + 1 11 Then it is not balanced!

We need to make the same change to the other side! A Question of Balance What happens if we change one of the sides of a balanced equation? 11 + 1 8 + 3 + 1 Then it is not balanced! We need to make the same change to the other side!

A Question of Balance 8 + 3 + 1 11 + 1 What happens if we change one of the sides of a balanced equation? 8 + 3 + 1 11 + 1 The 11th Commandment (for equations): We need to make the same change to the other side! We need to make the same change to the other side! Whatever thou dost unto the left, thou also must do unto the right.

To solve an equation means to find every number that makes the equation true. We do this by adding or subtracting to each side of the equation … but always keep it balanced!

Let’s go back to the balance In the equation, 7 added to a number gives 15… Solving the equation means, finding the value of the variable that makes the equation true. Let’s go back to the balance

x + 7 - 7 15 - 7 Simplify both sides Subtract 7 from both sides The 11th Commandment (for equations): Whatever thou dost unto the left, thou also must do unto the right. x + 7 - 7 15 - 7 Subtract 7 from both sides Simplify both sides

x 8 Subtract 7 from both sides Simplify both sides The 11th Commandment (for equations): Whatever thou dost unto the left, thou also must do unto the right. x 8 Subtract 7 from both sides Now we know the value of x Simplify both sides

The 11th Commandment (for equations): Whatever thou dost unto the left, thou also must do unto the right. x 8 So the solution goes like this… x + 7 = 15 x + 7 – 7 = 15 – 7 Subtract 7 from both sides Simplify both sides x = 8 Now we know the value of x

In some equations, the solution is obvious. x – 7 = 12 5n = 35 x = 19 n = 7 = 3 20 + h = 41 h = 21 c = 24 We can simply work the operation backwards in our head to get the answer.

But in other equations, the solution is not so obvious. We have to know what operation(s) must be done to solve it, and work it out carefully.

But in other equations, the solution is not so obvious. You have to do the inverse operation to both sides to get the variable by itself But in other equations, the solution is not so obvious. The opposite of addition is subtraction The opposite of subtraction is addition The opposite of multiplying by is multiplying by The opposite of multiplication is division

Multi-step equations When an equation has more than one operation you still have to isolate the variable by doing the following: Make sure variable terms are all on one side, and constant terms are on the other. Simplify Divide by the coefficient of the variable.

How would we solve 3x + 5 = 12? Let’s take another look at the balance 3x + 5 – 5 12 – 5 Subtract 5 from both sides

How would we solve 3x + 5 = 12? Let’s take another look at the balance 3x 7 Subtract 5 from both sides Simplify

Divide both sides by coefficient of the variable (3) How would we solve 3x + 5 = 12? Let’s take another look at the balance 3x 7 3 3 Subtract 5 from both sides Simplify Divide both sides by coefficient of the variable (3)

Divide both sides by coefficient of the variable (3) How would we solve 3x + 5 = 12? Let’s take another look at the balance 7 x 3 Subtract 5 from both sides Simplify So the solution is: Divide both sides by coefficient of the variable (3)

Let’s try some more equations Remember, we have to keep the equations balanced! Solve: 8m – 10 = 36 8m – 10 + 10 = 36 + 10 8m = 46 8 8 m = w = 84

5x  2 = x + 4 5x  2 + 2 = x + 4 + 2 5x = x + 6 5x – x = x – x + 6 Notice that there are variables on both sides Solve: 5x  2 = x + 4 5x  2 + 2 = x + 4 + 2 Get rid of the -2 on the left side 5x = x + 6 Simplify 5x – x = x – x + 6 Get rid of the x on the right side 4x = 6 Simplify Get rid of the cofficient of x 4 4 x = Simplify