Linear Equation Word Problem Translator
Objective
Slope recap
QUICK EXAMPLE:
SO, IF WE’RE GIVEN A GRAPH LIKE:
And we are asked to find the slope of this line, we know
we simply need to locate a point, and find how far away
the next point is away from it.
So, let’s take a look.
Well, if we start at the black point, and then count up…..
So, we can see that we needed to go up 3 units.
So, this means our rise for our slope is 3.
Now, we need to count how far to the left or right the
point is to really find the slope.�So, if we start at where we left off and count to the right
Slope Formula Recap
Find the Slope Examples
The Y-Intercept
After we mastered slope, we began to explore the y-intercept and what it had to offer to us when it comes to figuring out equations.
The y-intercept is where the graph crosses the y-axis (or when x = 0).
It also helps us find where the graph should start (again, it’s the point where the graph crosses the y-axis.)
HOW THE Y-INTERCEPT WORKS
SO THE WAY WE FIND THE Y INTERCEPT, IN A LINEAR EQUATION, IS BY LOOKING AT WHAT IS BEING ADDED TO THE X AND THE SLOPE.
SO, FOR EXAMPLE, IF WE HAVE AN EQUATION LIKE:
Y = 5 + 3X
WE KNOW THAT 5 IS THE Y-INTERCEPT.
WE KNOW THIS BECAUSE IT IS THE ONLY OTHER NUMBER IN THE EQUATION OTHER THAN THE SLOPE, AND IT IS BEING ADDED TO THE SLOPE.
(IT CAN ALSO BE SUBTRACTED, THAT JUST MAKES IT A NEGATIVE).
SO, IF WE HAD A GRAPH OF,
SAY THE EQUATION GIVEN,
IT WOULD LIKE SOMETHING LIKE:
How to find the equation given two points
So, let’s say we are given two points and asked to find the equation of the line.
There are a few things we must do.
FIRST - we find the slope of the line.
SECOND – we use the slope to find the y-intercept by plugging in a point.
THIRD – we CHECK OUR ANSWER by plugging in a point again.
Seems simple enough right? Well, let’s take a look at an example
Example
EXAMPLE CONTINUED
NOT DONE!
Translating word problems
Now let’s move to translating word problems for linear equations.
It’s true that everyone hates word problems
But that’s mostly because we don’t know how to do it.
Once we can see how we can translate word problems into equations, solving word problems can be really easy.
So, let’s go ahead and start:
Dissecting a word problem
So to start off, let’s look at a word problem, then take it apart to see what it all means.
Mike just developed an app for Apple, and wants to sell it. He contacts Real Engine, who offers him $5000 up front, and then $2.5 per download. Write a linear equation that would allow Mike to see how long he would have to wait to make $7000.
Alright, so what we want to look for is some way to figure out how to make this word problem into an equation.
To do that, we need to look for some key words like:
Per
Of
Is
Etc.
Well, first off, we can see that the first sentence has no useful information at all.
So:
Mike just developed an app for Apple, and wants to sell it. He contacts Real Engine, who offers him $5000 up front, and then $2.5 per download. Write a linear equation that would allow Mike to see how long he would have to wait to make $7000.
Now, let’s look towards the next sentence.
Well, we can see that we see the word per, which means the number right behind it is going to be the slope of the linear equation.
We know this because per is a keyword to look for.
So:
If y = mx + b
Where m is the slope
And b is the y-intercept
Then we know that:
y = 2.5x + b
So, the hard part is done.
Now we just need to find b
Well, there is only one other number in the sentence we’re looking at, which would be the $5000
Since we know 2.5 is the slope, then $5000 must be our y-intercept!
y = 2.5x + 5000
And finally, what they asked us to do is to find out when he would make $7000.
So, we take our equation
Make y = $7000
Then solve for x.
So:
7000 = 2.5x + 5000
And now we solve:
- 5000 - 5000
2000 = 2.5x
_____________
2.5 2.5
800 = x
And now we know that Mike needs to have 800 downloads of his new app to make $7000!
Now let’s try to look at a few more:
Example 2:
Now that we know more of what to do, let’s look at another word problem:
Jamil just inherited an apartment complex from his deceased grandfather. He ran the finances for the complex for years, and he knows that after bills and utilities, the complex makes an income of $20,453 per month. He really wants to buy a house of his own at some time, but he doesn’t want to take away from the apartment complex. If his grandfather left him $100,000 in a savings account, how long will he have to wait to afford a $400,000 house?
Again, to figure this out, all we want to do is look for our key words:
Per
Is
Etc.
Well, looking at the first sentence, we know it has nothing to do with the equation.
Jamil just inherited an apartment complex from his deceased grandfather. He ran the finances for the complex for years, and he knows that after bills and utilities, the complex makes an income of $20,453 per month. He really wants to buy a house of his own at some time, but he doesn’t want to take away from the apartment complex. If his grandfather left him $100,000 in a savings account, how long will he have to wait to afford a $400,000 house?
Now, let’s look towards the next sentence.
Well, we can see that we see the word per, which means the number right behind it is going to be the slope of the linear equation.
We know this because per is a keyword to look for.
So:
If y = mx + b
Where m is the slope
And b is the y-intercept
Then we know that:
y = 20,453x + b
So, the hard part is done.
Now we just need to find b
Well, we can see another number after the slope, which is $100,000.
Although it’s not in the same sentence, since it’s still what Jamil would have, then we can say this is the y-intercept since it’s the only other number.
So:
y = 20,453x + 100,000
So now, we have our equation.
Finally they want to know how long it will take for Jamil to make enough for a $400,000 house.
So, now we solve, setting y equal to 400,000
400,000 = 20,453x + 100,000
And now we solve:
- 100,000 - 100,000
300,000 = 20,453x
_________________
20,453 20,453
14.7 = x
So, this would mean that Jamil needs to wait 14.7 months to afford his house!
Last Example:
Now that we know more of what to do, let’s look at another word problem:
The water level in California is terrible, and they are in a drought. Researchers have seen that California has a surplus of only 300,000 gallons per year. They estimate that the amount of water needed to pull them out of this drought is 25 million gallons of water. If all of the reservoirs combined only have 4,000,000 gallons, how long until California will have enough water?
Again, to figure this out, all we want to do is look for our key words:
Per
Is
Etc.
Well, looking at the first sentence, we know it has nothing to do with the equation.
The water level in California is terrible, and they are in a drought. Researchers have seen that California has a surplus of only 300,000 gallons per year. They estimate that the amount of water needed to pull them out of this drought is 25 million gallons of water. If all of the reservoirs combined only have 4,000,000 gallons, how long until California will have enough water?
Now, let’s look towards the next sentence.
Well, we can see that we see the word per, which means the number right behind it is going to be the slope of the linear equation.
We know this because per is a keyword to look for.
So:
If y = mx + b
Where m is the slope
And b is the y-intercept
Then we know that:
y = 300,000x + b
So, the hard part is done.
Now we just need to find b, which gets tricky.
Technically, the next number we see is 25 million.
But that’s not what California has, but what it needs.
So what do they have?
Well, the other number we see is 4,000,000.
Again, even though it’s way later, it’s what California has. So this would count as our b.
y = 300,000x + 4,000,000
So now, we have our equation.
And, like we saw first, California needs to make 25 million gallons of water to be set for a while.
So, if we want to know when they will reach it, we need to let y = 25,000,000
So:
25,000,000 = 300,000x + 4,000,000
And now we solve:
- 4,000,000 - 4,000,000
21,000,000 = 300,000x
_______________________
300,000 300,000
70 = x
So, at this rate, it’ll take California 70 years to finally replenish what they need to pull out of the drought.
Sounds about right.
SO THAT’S HOW WE DO IT
Again, first thing we look for is the word “per” (or something that seems like “per”).
And we set the number next to it as our slope.
Then we look for our y – intercept, which will be the number that we have (not the one we need).
Once our equation is set up, we set y equal to what we need.
Then solve!