MPM 1DI ­ U3 ­ D7 ­ Modelling with Formulas.notebook
November 08, 2016
MPM 1DI
Unit 3 ­ Equations
Day 7 ­ Modelling with Formulas
Today's learning goal: Rearranging equations to solve for a given variable
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MPM 1DI ­ U3 ­ D7 ­ Modelling with Formulas.notebook
November 08, 2016
Unit 3 ­ Equations Modelling with Formulas
Today's learning goal: Rearranging equations to solve for a given variable
How do I isolate x (or P or T...) in a formula? Sometimes, it may seem like mathematicians are speaking another language when we talk about equations or formulae. y=mx+b, E=mc2, C=2πr, P=2(l+w), etc...
Especially if they expect you to "manipulate" or rearrange them! But, equations can provide powerful tools for describing the natural world. With a little algebra, we can rearrange those equations to solve for ANY of the variables in them. Although this may seem like magic, you don't have to be a "mathemagician" to do this. Today we will work on some tools to call upon to help you to learn some simple steps to help you to solve an equation for any of the variables.
Why should I manipulate equations?
Believe it or not, there are many good reasons to develop your ability to rearrange equations that are important to the many disciplines. 1.
It can save time, help you with units and save some brain space! 2.
Equations are easier to handle before inserting numbers! And, if you can isolate a variable on one side of the equation, it is applicable to every similar problem that asks you to solve for that variable! 3.
If you know how to manipulate equations, you only have to remember one equation that has all the variables of question in it ­ you can manipulate it to solve for any other variable! This means less memorization! 2
MPM 1DI ­ U3 ­ D7 ­ Modelling with Formulas.notebook
November 08, 2016
Some Rules to Follow
RULE #1: you can add, subtract, multiply and divide by anything, as long as you do the same thing to both sides of the equals sign. ... Let's use the equation for a line to illustrate an example of how to use Rule #1. The general equation for a line is: y = mx +b
• If we wish to solve for b in this equation, we must subtract mx from both sides. • If we perform the math on each side (that is, subtract mx from mx on the right), we end up with an equation that looks like this: This equation can also be written ______________, if you prefer to have the solved variable on the left.
RULE #2: to move or cancel a quantity or variable on one side of the equation, perform the "opposite" operation with it on both sides of the equation
Let's use the equation for Area of a circle to demonstrate Rule #2. The general equation for Area of a circle is: • If we wish to solve for r in this equation, we must ________ by ___on both sides. • If we perform the math on each side we end up with an equation that looks like this.
• We the need to isolate r by doing the opposite of squaring which is________ ___________
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MPM 1DI ­ U3 ­ D7 ­ Modelling with Formulas.notebook
November 08, 2016
Example 1: Rearrange the following equations for the indicated variable:
a) Given the equation d = a + b , express a in terms of b and d.
b) Rearrange C = 2πr to isolate r
Rearranging an equation with more than one step.
Example 2: The equation of a line relates y to x, m and b: y = mx + b. Rearrange this equation to express x in terms of y, m and b.
** Note we'll talk more about the significance of m and b in this equation in a later chapter!!!
Steps: 1. Isolate the variable term
2. Isolate the variable
Example 3: The kinetic energy of an object depends on its mass and how it is moving and can be modelled by the formula Where :
E is the kinetic energy (in joules)
m is the mass (in kilograms)
v is the speed (in metres per second)
If the mass of an object is 25 kilograms and it has a kinetic energy of 125 joules, what is the object's velocity?
1. Assess what you have (which of the variables do you have values for? DO NOT plug in any numbers yet! 2. Determine which of the variables you want as your answer. (What is the question asking you to calculate? What is the unknown variable?)
3. Rearrange the equation so that the unknown variable is by itself on one side of the equals sign (=) and all the other variables are on the other side. 4. NOW plug in the numbers! Replace known variables with their values and calculate using correct order of operations.
5. Ask yourself whether the answer is reasonable in the context of what you know.
Pg. 215-219 #1-3, 6-8, 10-12, 15, 16a
Challenge Page 219 #18, 19
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