Linear Equations 101: Math Every Algebra Student Should Know
You've probably been to a theme park like Disneyland, or carnival at least once. What the heck does this have to do with linear equations? Buckle up and let me explain it to you.
Fun Day at the Carnival!
Imagine you and your parents are at the entrance of a local carnival.
You are at the ticket booth and the ticket seller tells you that parties (groups) larger than two people cost $40.
You pay the ticket seller $40 just to get inside of the carnival venue. Wow! There are over 50 exciting game booths so you walk up to one of the employees and ask how much each game costs. He/she tells you that each game inside of the carnival is $5.
Now, like any child does, you ask your mother for a $5 bill to try and win a prize from the game booths.
Since the basketball booth is the first station that catches your eye, you quickly sprint to the attendant and give the $5 bill to them. You take the basketball and shoot it.
OOPS! You missed (I can never shoot a basketball)! So, you decide to try again with another $5 bill.
OOPS! You missed AGAIN!
You give up and proceed to roam around the carnival and by the end of the night, you end up playing 72 games. Whew! That's a lot of games!
However, at the end of the day, since your mom paid with cash instead of a credit or debit card, she has no digital record of how much she spent.
So. . . how can she calculate how much she spent?
This is where the concept of linear equations come in.
Before I tell you how linear equations has anything to do with the scenario above, it is crucial that you understand the basics of linear equations. By basic, I mean BASIC!
There are 3 ways to write linear equations:
Slope-Intercept Form: y = mx + b
Point-Slope Form: y - y1 = m ( x - x1 )
Standard Form: ax + by = c
We'll mainly be covering how the scenario is most applicable to Slope-Intercept Form of a line. I'll explain what Point-Slope form and Standard Form in another post. Let's get to it.
In Slope-Intercept Form, equations are written in the form of y = mx + b.
The "x" is known as the independent variable, meaning that it can change without depending on anything.
The "y" is affected by the "x", and because of this, it's known as the dependent variable. "Y" cannot be whatever number and can't change randomly like "x" can; it relies on "x" for a specific answer known as an output.
y = the total value of everything after "x" is plugged in.
x = slope or the rate of change (Ex: Dollars per week, cost per object, etc.)
b = b is what the "y" value equals when you plug "x" for 0. This is also known as the Y-Intercept.
The Y-intercept is where (usually written in a coordinate) the line crosses or intersects the Y-axis.
Now, let's summarize the scenario I just described.
Paid $40 for entry fee
$5 per game
You played a total of 72 games
Ask yourself this question: What are my independent and dependent variables?
Your answer should be: The number of games you play is the independent variable (x-variable) since you can play as many times as you want.
The dependent variable (y-variable) is the total amount of money spent since it depends on how many games you play.
Think about it, when you paid the entry fee of $40, you haven't even played any games at all. In other words, you played 0 games when you entered the carnival. Remember, the Y-intercept is represented by the variable "b" in our equation y = mx + b. It's when our x-value is equal to 0.
Therefore, $40 is our Y-intercept or "b." Now we can start slowly building our equation.
y = mx + (40)
Moving on, we know that it cost $5 to play each game. That means if I wanted to play 3 games, it would cost me $15 (5 x 3 = 15), and if I wanted to play 10 games, it would cost me $50 (5 x 10 = 50). Since no matter how many games I play, the amount of money I need per game will always be the same, this is known as my slope or rate of change. Remember, the slope is represented by the "m" variable.
Hence, $5 is our slope or rate of change, and can be plugged into the equation for the variable "m." y = (5)x + (40)
**Note that when writing equations, you leave "x" and the "y" untouched since you will be plugging in values in for either "x" or "y."
We finally have an equation that will represent how much money it costs in total to play games at the carnival. y = 5x + 40
Since we now know that the number of games we played (72 games) is represented by the x-variable, we can substitute 72 for "x." After we solve it, we should get the total cost of going to the carnival and paying for the games.
y = 5 * (72) + 40 --> y = 360 + 60 --> y = 420.
The total value is represented by the "y" as stated previously. Therefore, we can conclude that $420 was spent at the carnival in total.
Try this problem yourself!
Allie and her mother eat at a restaurant that charges $5 per piece of tempura. Since many celebrities have been there, the restaurant charges $10 per person just to enter the restaurant. Unfortunately, the restaurant does not accept cash. Allie eats 17 pieces of tempura, and her mom eats 3 pieces of tempura. How much did it cost Allie and her mother altogether to eat at the restaurant?
$5 per piece of tempura (My slope, represented by m, is 5).
$10 per person ( Allie and her mother attended so 10 * 2 = $20). Therefore, $20 is my Y-intercept; in other words, they will need to pay $20 just to get into the restaurant.
Altogether, Allie and her mother ate a total of 20 pieces of tempura.
20 pieces is my "x" value that I plug in for "x" in my y = mx + b equation.
My Equation is: y = 5x + 20
I plug in 20 for "x" and get 5(20) + 20 which gives me a grand total of $120!
There you have it: the basics of linear equations.
With this concept under your belt, you're not only able to ace your tests, but you're also able to apply it to real-life scenarios to save you a TON of time!
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