Solve using a system of equations.

Question

Lucy and Greg each want to buy a bicycle. Lucy has already saved $35 and plans to save $10 per week. Greg has $26 and plans to save $13 per week. How many weeks will it take for Lucy and Gregory to have the same amount of money saved? How much money will they have saved?

Answer 1

First, we translate each verbal statement into a mathematical one:

First, Lucy:

"Lucy has saved $35 and plans to save $10 per week." Her initial $35 will only contribute to her total sum (S) at the beginning, so that doesn't get a variable, and stays as 35. Her $10 per week will continue to increase her total savings each week, so it gets multiplied by a variable, w, which represents the number of weeks: 10w

We combine S, 35, and 10w into a single statement showing that her total sum of money after w weeks will be equal to her initial $35 plus $10 per week: S = 35 + 10w.

Next, Greg:

"Greg has $26 and plans to save $13 per week."

His initial $26 will only contribute to his total sum (S) at the beginning, so that doesn't get a variable, and stays as 26.

His $13 per week will continue to increase his total savings each week, so it gets multiplied by a variable, w, which represents the number of weeks: 13w

(Note: we eventually will want to compare Greg and Lucy's savings, so we want to use all the same variables)

We combine S, 26, and 13w into a single expression showing that his total sum of money after w weeks will be equal to his initial $26 plus $13 per week: S = 26 + 13w.

Now, since we want to know after how many weeks Lucy and Greg will have the same amount of money, we know that S in each equation will be the same amount, so we know that each expression (S = 26 + 13w and S = 35 + 10w) will equal each other, and we can combine them all into a single expression: S = 26 + 13w = S = 35 + 10w.

For now, we're just interested in finding out what w is, since the question is asking how many weeks it will take for Greg and Lucy to have the same amount of money, so we can remove S from that expression:

26 + 13w = 35 + 10w (we know that these equal each other because we know that they both equal S)

Now we solve for w:

Now, we want to isolate all the terms with w on one side of the equation. We can subtract 26 from each side:

26 - 26 + 13w = 35 - 26 + 10w

13w = 9 + 10w

Now we subtract 10w from each side:

13w - 10w = 9 + 10w - 10w

3w = 9 (Reminder: when performing subtraction (or addition) with terms that each has the same variables (in this case subtracting 10w from 13w), we add or subtract the numbers and the variable is unchanged. In this case, the w disappeared from the right side because 10w - 10w = 0w = 0, because any number times 0 is 0.)

Finally, we divide each side by 3 to isolate w:

3w ÷ 3 = 9 ÷ 3

w = 3 weeks

Now we know that it will take 3 weeks for Greg and Lucy to have the same amount of money! But, we are not done yet, because the question also asked what that amount is. To figure that out, we take either one of those equations (S = 26 + 13w and S = 35 + 10w) and plug in 3 for w (we can use either equation because we know they are both equal):

S = 26 + 13 × 3

S = 26 + 39

S = 65

Just to make sure that we're correct, let's try it with Lucy's equation:

S = 35 + 10w

S = 35 + 10 × 3

S = 35 + 30

S = 65

Answer 2

Hello, Eirlys,

I use "L" as Lilly's total money saved, and "G" for Greg's amount saved. Let x be the number of weeks.

We want to know when Lilly and Greg have equal amount of money, which we can state as:

L = G

We can write equations for both by taking their starting savings and adding their earnings, which would be 10x for Lilly and 13x for Greg:

L = 35 + 10x

G = 26 + 13x

We want to know the value of x when both have the same amounts. Just use the L=G to state:

35+10x = 26+13x

Now we solve for x:

9 = 3x

x = 3

Check the answer:

L = 35 + 10*3 = $65

G = 26 + 13*3 = $65

It will take 3 weeks for their savings to be equal.

Bob