Problem 62 - College Algebra

Question

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 numbers are 7 and -2.

Step 1: Identify the Variables

Let's set up our variables for the two unknown numbers we want to find. Let \( x \) be the first number, and \( y \) be the second number. Thus, our task is to find the values of \( x \) and \( y \).

Step 2: Translate the Word Problem into Equations

We have two conditions from the problem: 1. A number is 9 more than another number. This translates to the equation \( x = y + 9 \).2. Twice the sum of the two numbers is 10. This translates to \( 2(x + y) = 10 \).

Step 3: Simplify the Second Equation

Start by simplifying the second equation: \[ 2(x + y) = 10 \]Divide both sides by 2 to obtain:\[ x + y = 5 \]

Step 4: Substitute to Solve for One Variable

Use the equation from Step 2: \( x = y + 9 \). Substitute \( x \) in the simplified equation \( x + y = 5 \) from Step 3:\[ (y + 9) + y = 5 \]

Step 5: Solve for y

Simplify and solve the equation from Step 4:\[ 2y + 9 = 5 \]Subtract 9 from both sides:\[ 2y = -4 \]Divide by 2 to solve for \( y \): \[ y = -2 \]

Step 6: Substitute y to Find x

Now use the value of \( y = -2 \) in the equation \( x = y + 9 \):\[ x = (-2) + 9 \]Solve for \( x \):\[ x = 7 \]

Step 7: Verify the Solution

Finally, check if \( x = 7 \) and \( y = -2 \) satisfy both equations.1. \( x = y + 9 \) gives \( 7 = -2 + 9 \) which is true.2. \( 2(x + y) = 10 \) gives \( 2(7 + (-2)) = 10 \), or \( 2(5) = 10 \), which is also true.

Key Concepts

Two Variables

When solving a system of linear equations, it's crucial to define the variables clearly. In this exercise, we have two unknowns, so we introduce two variables to represent them. We can use any letters, but common choices are \( x \) and \( y \).

• \( x \) represents the first number.
• \( y \) represents the second number.

Using two variables allows us to set up equations based on the relationships described in the problem. Each variable will correspond to a specific equation, reflecting the story the problem tells us.
For example, the sentence 'A number is 9 more than another number' becomes \( x = y + 9 \), linking the variables directly. It's important to ensure that the equations accurately represent the problem scenario.

Equation Solving

Equation solving is the process of finding the values of the variables that satisfy the given equations. In this problem, we use two key equations derived from the word problem:

• \( x = y + 9 \)
• \( 2(x + y) = 10 \)

To solve these equations, we first simplify each one if possible. For instance, the second equation can be simplified by dividing by 2, resulting in \( x + y = 5 \).
The goal is to manipulate these equations so that one of the variables can be isolated. Often, this involves techniques like substitution or elimination. Each step must be done carefully to maintain the equality of the equation.

Substitution Method

The substitution method is a commonly used approach for solving systems of linear equations. It involves solving one equation for one variable and then substituting this expression into the other equation. This helps to eliminate one of the variables, simplifying the problem to a single equation with one unknown variable.
In our exercise, we isolate \( x \) in the equation \( x = y + 9 \). Then we substitute \( y + 9 \) in place of \( x \) in the other equation \( x + y = 5 \):
\[(y + 9) + y = 5\]This replacement turns our problem into a single variable equation \( 2y + 9 = 5 \), which simplifies to \( y = -2 \).
After solving for \( y \), replace \( y \) back into the initial expression to find \( x \). This method is effective for equations where one variable can be easily expressed in terms of the other, making it a handy tool in systems of equations.