In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

Verified step by step guidance

1. Define the variables: let $x$ represent one number and $y$ represent the other number.
2. Translate the first condition 'The sum of two numbers is 7' into an equation: $x+y=\; 7$.
3. Translate the second condition 'If one number is subtracted from the other, their difference is -1' into an equation. This can be written as $x-y=\; -1$.
4. Set up the system of equations: $\backslash begin\{cases\}\; x\; +\; y\; =\; 7\; \backslash \backslash \; x\; -\; y\; =\; -1\; \backslash end\{cases\}$.
5. Solve the system by adding the two equations to eliminate $y$: $(x\; +\; y)\; +\; (x\; -\; y)\; =\; 7\; +\; (-1)$. Simplify and solve for $x$, then substitute back to find $y$.

Key Concepts

Formulating Systems of Equations

This involves translating word problems into mathematical equations using variables. Here, x and y represent the two numbers, and conditions like their sum and difference are expressed as equations. Accurate formulation is essential for solving the problem.

Solving Systems of Linear Equations

Once the system is set up, methods such as substitution or elimination are used to find the values of variables that satisfy both equations simultaneously. This process yields the specific numbers that meet the given conditions.

Interpreting Solutions in Context

After solving the system, the numerical solutions must be interpreted in the context of the problem. This ensures the answers make sense and correspond to the original quantities described, confirming the problem is correctly solved.