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. Step 1: Let's denote the two numbers as x and y. The problem gives us two conditions that we can translate into equations. The first condition is that the sum of the two numbers is 7. We can write this as an equation: x + y = 7.
2. Step 2: The second condition is that if one number is subtracted from the other, their difference is -1. We can write this as another equation: x - y = -1.
3. Step 3: Now we have a system of two equations: x + y = 7 and x - y = -1. We can solve this system of equations using either substitution or elimination method.
4. Step 4: If we use the elimination method, we can add the two equations together. The y terms will cancel out, and we can solve for x.
5. Step 5: Once we have the value of x, we can substitute it into one of the original equations to solve for y.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. In this context, we need to create two equations based on the relationships described in the problem. Solving the system involves finding values for the variables that satisfy all equations simultaneously.

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines. Each equation in the system can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding how to manipulate and solve these equations is crucial for finding the values of x and y.

Substitution and Elimination Methods

Substitution and elimination are two common methods for solving systems of equations. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable.