Problem 62 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. A number is 9 more than another number. Twice the sum of the two numbers is 10 . Find the two numbers.

Step-by-Step Solution

Answer

The two numbers are 7 and -2.

Step 1: Define the Variables

Let's begin by defining our variables. Let \( x \) be the first number and \( y \) be the second number.

Step 2: Set Up the First Equation

According to the problem, the first number is 9 more than the second number. This gives us the equation: \( x = y + 9 \).

Step 3: Set Up the Second Equation

The problem states that twice the sum of the two numbers is 10. This means \( 2(x + y) = 10 \). Simplifying this, we get \( x + y = 5 \).

Step 4: Substitute the First Equation into the Second Equation

Substitute \( x = y + 9 \) from the first equation into \( x + y = 5 \). This results in \( (y + 9) + y = 5 \).

Step 5: Solve for y

Simplify and solve the equation \( y + 9 + y = 5 \). This becomes \( 2y + 9 = 5 \). Subtracting 9 from both sides, we have \( 2y = -4 \). Dividing both sides by 2 gives \( y = -2 \).

Step 6: Solve for x

Now that we have \( y = -2 \), substitute it back into the equation \( x = y + 9 \). This results in \( x = -2 + 9 \), so \( x = 7 \).

Step 7: Conclusion

The two numbers we found are \( x = 7 \) and \( y = -2 \). These numbers satisfy both original equations.

Key Concepts

Variables in Equations

In algebra, a variable is simply a symbol that represents a number. We use variables because they allow us to write general expressions and equations that describe relationships between numbers. In this exercise, we have two unknown numbers. To solve for these unknowns, we assign variables to them.

Typically, variables are represented using letters such as \( x \) and \( y \). In our problem, we let \( x \) be the first number and \( y \) the second. This is helpful because it switches our focus from specific numbers to the relationships between them.

• \( x \) denotes the first number
• \( y \) denotes the second number

By using variables, we can easily manipulate and rearrange equations to find the values of these unknown numbers. Understanding the role of variables is crucial in forming equations that can be solved to find unknown values.

Solving Equations

Solving equations involves finding the values that satisfy the equations. In the problem we're working with, we have two equations that involve two variables. Solving this system means finding the values of \( x \) and \( y \) that make both equations true.

Let's recap the main steps of solving equations:

• Start by defining the equations based on the relationships described in the problem. This gives you the mathematical structure needed to find solutions.
• We substitute variables from one equation into the other to streamline our system of equations into one where a single variable is solved.
• After substituting, solve the simplified equation. This involves combining like terms, moving terms to different sides of the equation, and performing operations like addition, subtraction, multiplication, or division to isolate the variable.

In our example, once we substituted \( x = y + 9 \) into \( x + y = 5 \), we were able to solve for \( y \). Then, using the value of \( y \), we calculated the value of \( x \). These steps unravel the solution to the system of equations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. Understanding them is key to setting up and solving equations. In the given exercise, two algebraic expressions are transformed into mathematical equations, which reflect the problem's statements.

Here's a breakdown of some of the algebraic expressions used:

• \( x = y + 9 \) is an expression indicating that the first number is 9 units more than the second number.
• \( 2(x + y) = 10 \) expresses that twice the sum of these numbers equals 10.
• When simplified, \( x + y = 5 \) tells us that the sum of the two numbers equals 5.

Algebraic expressions allow us to represent the relationships and constraints of a problem efficiently. They give us a way to communicate complex ideas simply and clearly with the use of symbols and numbers. They are foundational to writing the equations necessary for solving problems involving systems of linear equations. By becoming comfortable with crafting and manipulating algebraic expressions, you can better tackle a wide range of algebra problems.