Question

1. the sum of two numbers is 63. their difference is 19. write a linear system to model this problem and use it to find the two numbers.

Explanation:

Step1: Define Variables

Let the two numbers be \( x \) and \( y \), assume \( x > y \).

Step2: Write the Sum Equation

The sum of the two numbers is 63, so \( x + y = 63 \).

Step3: Write the Difference Equation

Their difference is 19, so \( x - y = 19 \).

Step4: Solve the System

Add the two equations: \( (x + y) + (x - y) = 63 + 19 \)
Simplify: \( 2x = 82 \), then \( x = \frac{82}{2} = 41 \).
Substitute \( x = 41 \) into \( x + y = 63 \): \( 41 + y = 63 \), so \( y = 63 - 41 = 22 \).

Answer:

The linear system is \(

$$\begin{cases} x + y = 63 \\ x - y = 19 \end{cases}$$

\), and the two numbers are 41 and 22.