Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Isaiah and his little sister are saving up money to buy a joint birthday present for their mother. Isaiah already has $\$24$ saved and plans to save $\$14$ per week from his allowance. His sister has $\$16$ saved so far and will save $\$18$ per week from hers. The two siblings will soon have saved the same amount towards their mother's gift. How long will that take? How much will each one have saved? $\newline$ In $\_\_\_$ weeks, Isaiah and his sister will each have saved $\$\_\_\_\_\_$ .

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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

Isaiah and his little sister are saving up money to buy a joint birthday present for their mother. Isaiah already has

\$24

saved and plans to save

\$14

per week from his allowance. His sister has

\$16

saved so far and will save

\$18

per week from hers. The two siblings will soon have saved the same amount towards their mother's gift. How long will that take? How much will each one have saved?

\newline

\_\_\_

weeks, Isaiah and his sister will each have saved

\$\_\_\_\_\_

Representation and Equations: Let $x$ represent the number of weeks, and $y$ represent the total amount saved. For Isaiah: Initial savings: $\$24$ , Weekly saving rate: $\$14$ . Equation: $y = 14x + 24$ .
Isaiah and Sister's Savings: For Isaiah's sister: Initial savings: \$$16$, Weekly saving rate: \$$18$. Equation: $y = 18x + 16$.
Solving for Same Amount Saved: System of equations: $y = 14x + 24$ and $y = 18x + 16$. Set the equations equal to find when they will have saved the same amount. $14x + 24 = 18x + 16$.
Finding the Value of $x$: Solve for $x$: Subtract $14x$ from both sides: $24 = 4x + 16$. Then subtract $16$ from both sides: $8 = 4x$. Divide by $4$: $x = 2$.
Substitution and Final Answer: Substitute $x = 2$ back into one of the original equations to find $y$. Using Isaiah's equation: $y = 14(2) + 24 = 28 + 24 = 52$.