Question

the sum of the ages of a brother and sister is 20. the sister is two years older than the brother. how old are the brother and sister?
let x represent the current age of the sister, and let y represent the current age of the brother.
which set of equations and method will work as a way to (eventually) solve for one variable in this problem?

equation 1: x + y = 20
equation 2: x = y + 2
subtract y from both sides of equation 1. use the method of substitution by setting up the equation y - 20 = y + 2.

equation 1: x + y = 20
equation 2: y = x + 2
subtract y from both sides of equation 2. use the method of elimination by subtracting the new equation 2 from equation 1.

equation 1: x + y = 20
equation 2: y = x + 2
divide both sides of equation 2 by 2. use the method of substitution by replacing y in equation 1 with \\(\frac{1}{2}x + 1\\).

equation 1: x + y = 20
equation 2: x = y + 2
subtract y from both sides of equation 2. use the method of elimination by adding the two equations together.

Explanation:

Step1: Verify equation setup

The problem states:

1. Sum of ages: $x + y = 20$ (correct for all options)
2. Sister is 2 years older: $x = y + 2$ (only first and fourth options have this correct; others incorrectly state $y = x + 2$)

Step2: Check first option logic

First option's substitution step: Subtract $y$ from Equation1 gives $x = 20 - y$, but it incorrectly sets $y - 20 = y + 2$, which is mathematically invalid.

Step3: Check fourth option logic

Fourth option: Subtract $y$ from Equation2: $x - y = 2$.
Add this new equation to Equation1:
$$(x + y) + (x - y) = 20 + 2$$
$$2x = 22$$
This eliminates $y$ and solves for $x$, which is valid elimination method.

Answer:

Equation 1: $x + y = 20$
Equation 2: $x = y + 2$
Subtract $y$ from both sides of Equation 2. Use the method of elimination by adding the two equations together.