Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mr. Ballard and Ms. Hong are teaching their classes how to write in cursive. Mr. Ballard has already taught his class $2$ letters. The students in Ms. Hong's class, who started the unit later, currently know how to write $12$ letters. Mr. Ballard plans to teach his class $5$ new letters per week, and Ms. Hong intends to cover $4$ new letters per week. Eventually, the students in both classes will know how to write the same number of letters. How long will that take? How many letters will the students know? $\newline$ In $\_$ weeks, the students in both classes will know how to write $\_$ letters in cursive.

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Mr. Ballard and Ms. Hong are teaching their classes how to write in cursive. Mr. Ballard has already taught his class

2

letters. The students in Ms. Hong's class, who started the unit later, currently know how to write

12

letters. Mr. Ballard plans to teach his class

5

new letters per week, and Ms. Hong intends to cover

4

new letters per week. Eventually, the students in both classes will know how to write the same number of letters. How long will that take? How many letters will the students know?

\newline

\_

weeks, the students in both classes will know how to write

\_

letters in cursive.

Define Variables: Let's define the variables: $\newline$ Let $x$ represent the number of weeks. $\newline$ Let $y$ represent the total number of letters learned by the students. $\newline$ For Mr. Ballard's class: $\newline$ Initial letters learned: $2$ $\newline$ Letters learned per week: $5$ $\newline$ Total letters learned over time: $y = 5x + 2$
Mr. Ballard's Class: For Ms. Hong's class: $\newline$ Initial letters learned: $12$ $\newline$ Letters learned per week: $4$ $\newline$ Total letters learned over time: $y = 4x + 12$
Ms. Hong's Class: Now we have two equations: $\newline$ $1$ ) $y = 5x + 2$ (Mr. Ballard's class) $\newline$ $2$ ) $y = 4x + 12$ (Ms. Hong's class) $\newline$ We will use substitution to solve for $x$ by setting the two equations equal to each other since $y$ represents the same total number of letters learned in both classes. $\newline$ $5x + 2 = 4x + 12$
Use Substitution: Solve for $x$ : $5x + 2 = 4x + 12$ $5x - 4x = 12 - 2$ $x = 10$ So, it will take $10$ weeks for the students in both classes to know the same number of letters.
Solve for x: Now we need to find out how many letters that will be. We can substitute $x = 10$ into either of the original equations. Let's use the first equation: $\newline$ $y = 5x + 2$ $\newline$ $y = 5(10) + 2$ $\newline$ $y = 50 + 2$ $\newline$ $y = 52$ $\newline$ Therefore, after $10$ weeks, the students in both classes will know how to write $52$ letters in cursive.