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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineIsaiah and his little sister are saving up money to buy a joint birthday present for their mother. Isaiah already has $24\$24 saved and plans to save $14\$14 per week from his allowance. His sister has $16\$16 saved so far and will save $18\$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?\newlineIn ___\_\_\_ weeks, Isaiah and his sister will each have saved $_____\$\_\_\_\_\_.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineIsaiah and his little sister are saving up money to buy a joint birthday present for their mother. Isaiah already has $24\$24 saved and plans to save $14\$14 per week from his allowance. His sister has $16\$16 saved so far and will save $18\$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?\newlineIn ___\_\_\_ weeks, Isaiah and his sister will each have saved $_____\$\_\_\_\_\_.
  1. Representation and Equations: Let xx represent the number of weeks, and yy represent the total amount saved. For Isaiah: Initial savings: $24\$24, Weekly saving rate: $14\$14. Equation: y=14x+24y = 14x + 24.
  2. Isaiah and Sister's Savings: For Isaiah's sister: Initial savings: \$\(16\), Weekly saving rate: \$\(18\). Equation: \(y = 18x + 16\).
  3. 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. \(14x + 24 = 18x + 16\).
  4. 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 \(4\): \(x = 2\).
  5. Substitution and Final Answer: Substitute \(x = 2\) back into one of the original equations to find \(y\). Using Isaiah's equation: \(y = 14(2) + 24 = 28 + 24 = 52\).

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