Using Formulas

Lesson 3: Writing and Rewriting Equations and Formulas

Oh boy, here we go

Writing Our Own Equations

When writing a linear equation to solve for a real world problem, we often use the slope-intercept form. the variable y to indicate the value we want and x to indicate the number of days, months, visitors, etc. that we expect.

*(The slope-intercept form is y = mx +b)

Writing Our Own Equations

The coefficients of the x variable will indicate how many dollars, sales, or pieces are expected per day, month, week, etc. The constant, b (called the y-intercept) is what we start with.

NOTE

When writing formulas, we write them in the form in which they are familiar. For instance, if I write a linear equation, then I write it in the form y = mx + b with each value in the space that it is represented by in the form.

Example Problems

• Marissa works at her job for $8 an hour. She starts this job with $1,000 in the bank already. How much money will she have if she works 20 hours her first week?

• Matt has to sell 40 teddy bears in order to reach his quota for the fundraiser. He has sold 12 teddy bears already. How many teddy bears does he need to sell in order to reach his quota?

• The trip from Birmingham to Mobile is about 300 miles. Pat has already driven 250 miles at an average speed of 50 mph. How many more hours does Pat have to drive if he keeps the same average speed?

Equations

All of these problems can be solved in one of two ways: reason it out or make an equation. Given the nature of the questions (the constant wage rate for Marissa, the consistent integrity of selling teddy bears, and the steadiness of Pat's speed), we can assume the resulting equation will be linear in form, not changing rates across time.

Let's Break Down Linear Equations Again

• Y is the result you are looking for, or are given in the problem.

• M is the rate at which the result changes over time.

• X is the time or quantity which has passed or been provided in the question.

• B is the starting point, or what they have with no work or time.

Example Problems: Write the Equation

• Marissa works at her job for $8 an hour. She starts this job with $1,000 in the bank already. How much money will she have if she works 20 hours her first week?

• Matt has to sell 40 teddy bears in order to reach his quota for the fundraiser. He has sold 12 teddy bears already. How many teddy bears does he need to sell in order to reach his quota?

• The trip from Birmingham to Mobile is about 300 miles. Pat has already driven 250 miles at an average speed of 50 mph. How many more hours does Pat have to drive if he keeps the same average speed?

Fill in the Blank

Clarence started his new job last week with $500 in the bank. Now, he has $800 in the bank after his first paycheck. He logged 40 hours at work during that first pay period. Write an equation to solve for his hourly wage.

Fill in the Blank

Solve the equation for Clarence's hourly wage.

Rewriting Formulas and Equations

How can we figure out different values using formulas we know?

Sample Formulas

• Find the length of the hypotenuse of the triangle ( $a^2+b^2=c^2$ )

• Find the area of the triangle          ( $A=\frac{1}{2}bh$ )

But could you solve for the height if given the area and base?

How to derive a formula from a formula

Creating a Formula from a Formula

• The formula for the area of a triangle is  $A=\frac{1}{2}bh$  

• But if we want to know the height of the triangle and are given the base and area, then we can isolate h in order to solve for the height of a triangle. So, we can perform the following steps to isolate h.

• Start:  $A=\frac{1}{2}bh$  

• Step 1. Multiply each side by 2 to cancel the 1/2 attached to h:  $2\cdot A=2\cdot\frac{1}{2}bh$  

• Step 2. Divide each side by b to cancel the b attached to h:  $\frac{2A}{b}=\frac{bh}{b}$  

• And we have derived the formula for height from the formula for area:  $\frac{2A}{b}=h$  

Why Derive These Formulas?

Because these formulas help us perform the same operation many times in a shorter span of time. Instead of plugging in to the area formula and solving for the missing variable every time, we can just plug in the values we are given into a derived formula and perform simple operations. Many physics formulas have been derived from previously discovered or derived formulas.

Example Problems

• Derive the formula for the height of a rectangle from the formula for the area of a rectangle:  $A=bh$  

• Derive the formula for the rate of an object using the distance formula:  $d=rt$  

• Derive the formula for the radius of a circle using the formula for the area of a circle:  $A=\pi r^2$  

• Derive the formula for the width of a rectangle from the perimeter of a rectangle formula:  $P=2l+2w$  

Multiple Choice

 Derive the formula for the diameter of a circle given the formula of the circumference:  $$C=\pi d$$  

• $$C=\frac{d}{\pi}$$  

• $$d=\frac{\pi}{C}$$  

• $$\frac{C}{\pi}=d$$  

• $$\pi C=d$$  

Multiple Choice

Derive the formula for the base of a triangle using the formula for the area of a triangle: $$A=\frac{1}{2}bh$$  

• $$\frac{2A}{b}=h$$  

• $$\frac{2A}{h}=b$$  

• $$\frac{\frac{1}{2}h}{A}=b$$  

• $$\frac{A}{2h}=b$$  

Fill in the Blank

Use your derived formula for the base of a triangle to solve for the base of a triangle with a height of 12 cm and an area of 96 cm².