If Amy multiplies her number by 2 and adds 3 times Aaron's number, the result is 18. Form another equation and hence solve them simultaneously to find the two numbers.

Let Amy's number be x and Aaron's number be y , and form the first equation: 2 x + 3 y = 18 .
Assume Amy's number is twice Aaron's number, creating the second equation: x = 2 y .
Substitute x = 2 y into the first equation and solve for y : y = 7 18 ​ .
Substitute the value of y back into x = 2 y to find x : x = 7 36 ​ , so the final answer is x = 7 36 ​ , y = 7 18 ​ ​ .

Explanation

Problem Introduction Let's break down this problem step by step to find Amy's and Aaron's numbers.

Forming the First Equation Let Amy's number be x and Aaron's number be y . We are given that if Amy multiplies her number by 2 and adds 3 times Aaron's number, the result is 18. This can be written as the equation: 2 x + 3 y = 18

Forming the Second Equation To solve for two variables, we need two independent equations. The problem only gives us one equation. To create a second equation, let's make a reasonable assumption. Let's assume that Amy's number is twice Aaron's number. This gives us the second equation: x = 2 y

Solving the System of Equations Now we have a system of two equations:


2 x + 3 y = 18 x = 2 y
We can use the substitution method to solve this system. Substitute the second equation into the first equation:

Substitution Substitute x = 2 y into the first equation: 2 ( 2 y ) + 3 y = 18

Solving for y Simplify and solve for y :
4 y + 3 y = 18 7 y = 18 y = 7 18 ​

Solving for x Now that we have the value of y , we can find the value of x using the second equation: x = 2 y = 2 ( 7 18 ​ ) = 7 36 ​

Final Answer So, Amy's number is 7 36 ​ and Aaron's number is 7 18 ​ .


Examples
Imagine you're baking cookies and need to adjust a recipe. If you know the total amount of flour and sugar should be 18 cups, and you want to use twice as much flour as sugar, you can set up a system of equations similar to the one we solved. By solving the equations, you can determine the exact amounts of flour and sugar needed to maintain the correct ratio and total quantity, ensuring your cookies turn out perfectly. This applies to various scenarios where you need to balance quantities based on given constraints and ratios.

Answered by GinnyAnswer | 2025-07-03

We defined Amy's number as x and Aaron's number as y . From the given conditions, we formed two equations: 2 x + 3 y = 18 and x = 2 y . Solving these equations, we found Amy's number to be 7 36 ​ and Aaron's number to be 7 18 ​ .
;

Answered by Anonymous | 2025-07-04