Question

1. your friend brian looks at this problem and says “you don’t have to bother graphing this, there’s no solution. john and brenda both save 20% of their income each month, which means their lines will have the same slope. if you graph them, they’ll be parallel.” is brian correct? why or why not?

1. john and brenda both want to have $25,000 saved, so they can be ready to pay for a down - payment on a house.

a. after how many months will john have met his savings goal?
b. after how many months will brenda have met her savings goal?
c. how can you use the graph to find those points?

1. brenda is considering increasing how much she saves each month, so she can meet her goal sooner.

a. how much would she need to save each month to reach $25,000 at the same time as john? hint: you can use the desmos graph from question 6 to help answer this question.
b. how did you find the answer to part a? describe your process or show your work

Explanation:

Step1: Assume John and Brenda's monthly savings amounts

Let's assume John's monthly income is $J_{income}$ and Brenda's monthly income is $B_{income}$. Their monthly savings are $0.2J_{income}$ and $0.2B_{income}$ respectively. Let the number of months be $m$, and the total savings $S$. The savings - equations are $S_J = 0.2J_{income}\times m$ and $S_B=0.2B_{income}\times m$. These are linear equations of the form $y = kx$ where $y$ is the total savings, $k$ is the monthly - savings amount (slope), and $x$ is the number of months. Since the percentage of income saved is the same (20%), the slopes of their savings - lines are equal if we consider the relationship between total savings and number of months.

Step2: Answer question 5

Brian is correct. In the context of linear equations representing their savings over time, if we consider the equations for their total savings $S$ as a function of the number of months $m$ (where $S =$ monthly savings $\times m$), since they both save 20% of their income each month, the slopes of their linear - savings functions are the same. Parallel lines have the same slope, and if the lines are parallel (and not the same line), there is no solution to the system of equations representing their savings in terms of finding a point of intersection (i.e., a time when they have the same non - general amount of savings).

Step3: Answer question 6a

Let John's monthly savings be $J_{monthly}$. We know that $S = J_{monthly}\times m$. If $S = 25000$, then $m=\frac{25000}{J_{monthly}}$. But we need to know John's monthly savings amount. If we assume John's monthly income is $I_J$, then $J_{monthly}=0.2I_J$. Without knowing $I_J$, we can't give a numerical answer. Let's assume for simplicity that we know John's monthly savings $J_{monthly}=x$. Then $m=\frac{25000}{x}$.

Step4: Answer question 6b

Similarly for Brenda, let Brenda's monthly savings be $B_{monthly}$. If $S = 25000$, then $m=\frac{25000}{B_{monthly}}$. If Brenda's monthly income is $I_B$, then $B_{monthly}=0.2I_B$. Without knowing $I_B$, we can't give a numerical answer. Let's assume Brenda's monthly savings is $y$, then $m = \frac{25000}{y}$.

Step5: Answer question 6c

On a graph with the number of months on the $x$ - axis and the total savings on the $y$ - axis, we would find the point where the $y$ - value (total savings) is 25000 for each of their lines. The $x$ - value at that point on John's line would be the number of months for John to reach the goal, and the $x$ - value at the point where $y = 25000$ on Brenda's line would be the number of months for Brenda to reach the goal.

Step6: Answer question 7a

First, find the number of months $m_J$ it takes John to reach $25000$. Then, let the new monthly savings amount for Brenda be $B_{new}$. We know that $25000=B_{new}\times m_J$. So $B_{new}=\frac{25000}{m_J}$.

Step7: Answer question 7b

We first find the number of months $m_J$ it takes John to reach $25000$ from his savings equation (if we know his monthly savings amount). Then, since Brenda wants to reach $25000$ in the same number of months $m_J$, we use the formula $B_{new}=\frac{25000}{m_J}$, where $B_{new}$ is the new monthly - savings amount for Brenda.

Answer:

1. Brian is correct. Their savings - lines are parallel as they have the same slope (20% of income saved per month), resulting in no solution (no intersection point) for non - identical lines.

6a. Without knowing John's monthly savings amount, we can't give a numerical answer. The formula is $m=\frac{25000}{J_{monthly}}$ where $J_{monthly}$ is John's monthly savings.
6b. Without knowing Brenda's monthly savings amount, we can't give a numerical answer. The formula is $m=\frac{25000}{B_{monthly}}$ where $B_{monthly}$ is Brenda's monthly savings.
6c. Find the point on each line where the $y$ - value (total savings) is 25000. The corresponding $x$ - value is the number of months.
7a. $B_{new}=\frac{25000}{m_J}$ where $m_J$ is the number of months it takes John to reach 25000.
7b. First find $m_J$ (number of months for John to reach 25000), then use $B_{new}=\frac{25000}{m_J}$ to find Brenda's new monthly savings amount.