Graph the system of equations from the previous subexercise. Is there a solution to the system?
A
y=0.5x+2.5 & (I) y=0.4x+5.8 & (II)
B
Yes, in the 33rd month.
Practice makes perfect
We want to write a system of linear equations that represents our friend's and her cousin's hair lengths over time. Let's begin by using a verbal model to write the system of linear equations. We will use x to represent the number of months and y to represent the hair length.
Notice that we are missing some information to complete this equation. We can use information from the exercise to find them. Let's look at the given table.
Month
Friend's Hair (in.)
Cousin's Hair (in.)
March
4
7
August
6.5
9
First, let's find how long our each girl's hair grows each month. We can subtract the hair length in March from the hair length in August and divide the result by the number of months that passed between them. This will give us the rate of hair growth per month.
Growth per month = Difference in length/Number of months
We know that March is the 3rd month of the year and August is the 8th month. This means that 8-3= 5 months have passed since the girls started measuring their hair. Now we have enough information to calculate the hair growth rates! Let's begin with our friend.
6.5-4/5=2.5/5=0.5
Now we will do the same to find the hair growth per month for our friend's cousin.
9-7/5=2/5=0.4
Notice that we still don't know the starting hair lengths for either our friend or her cousin. Let's call these values v and w, respectively. To find them, we will use the formulas for monthly hair growth that we created. We are told to use January as the starting month and, for simplicity, let's use March as the other month. We will begin with our friend.
We found that our friend's hair started at 2.5 inches. At the same time, her cousin's hair was 5.8 inches long. We finally have enough information to create a system of linear equations!
y=0.5x+2.5 & (I) y=0.4x+5.8 & (II)
We want to find out if our friend will be able to grow hair as long as her cousin's. We need to solve our system of equations!
y=0.5x+2.5 & (I) y=0.4x+5.8 & (II)
Let's graph each equation and look for a point of intersection, which will be the solution to the system of equations.
The graphs seem to intersect at the point (33,19). Finally, let's check if this point is a solution to the system of equations we created by substituting 33 for x and 19 for y in each equation and checking if they produce true statements.
