Problem 29 - Algebra A Combined Function

Question

For Exercises 29 and 30 , the solutions have been started for you. The first few exercises are each modeled by a system of two linear equations in two variables. One number is two more than a second number. Twice the first is 4 less than 3 times the second. Find the numbers. 1\. UNDERSTAND the problem. Since we are looking for two numbers, let \(x=\) one number \(y=\) second number 2\. TRANSLATE. Since we have assigned two variables, we will translate the facts into two equations. (Fill in the blanks.) 3\. SOLVE the system and 4\. INTERPRET the results.

Step-by-Step Solution

Answer

The numbers are 10 and 8.

Step 1: Understanding the Problem

We need to find two numbers. Let's denote these numbers as \(x\) and \(y\). One number \(x\) is two more than a second number \(y\), and twice \(x\) is 4 less than three times \(y\).

Step 2: Translating Words to Equations

Based on the problem statement, we can write two equations. The first equation is \(x = y + 2\) because the first number is 2 more than the second number. The second equation is \(2x = 3y - 4\) because twice the first number is 4 less than three times the second number.

Step 3: Solving the System of Equations

To solve the system, we first substitute \(x = y + 2\) from the first equation into the second equation: \(2(y + 2) = 3y - 4\). Simplifying this gives us: \(2y + 4 = 3y - 4\). Rearrange to get \(2y - 3y = -4 - 4\), which simplifies to \(-y = -8\). Thus, \(y = 8\). Substitute \(y = 8\) back into \(x = y + 2\) to find \(x\): \(x = 8 + 2 = 10\).

Step 4: Interpret the Results

The solution to the system shows that the first number \(x\) is 10, and the second number \(y\) is 8.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra. They are equations that represent straight lines when graphed on a coordinate plane. A linear equation in two variables generally looks like this: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables. These equations are called "linear" because they form a straight line when plotted. In this exercise, we have two linear equations: \(x = y + 2\) and \(2x = 3y - 4\). Together, these equations form a system that can be solved to find specific values of \(x\) and \(y\).

Two Variables

When dealing with linear equations, having two variables means there are two unknowns that we need to solve for. In our problem, the variables are \(x\) and \(y\), representing two numbers we need to find. The goal is to find values for both variables that satisfy the given equations. Often, systems of equations in two variables represent problems involving relationships between two quantities that depend on each other. This often appears in word problems where translating the problem into equations helps to find the relationships between different factors.

Solving Systems of Equations

Solving systems of equations means finding the values of the variables that fulfill all the equations in the system simultaneously. In this example, we have a system consisting of two equations. There are several methods to solve systems of equations, including:

• Graphical Method: Plotting both equations on a graph and finding the point of intersection, which represents the solution.
• Substitution Method: Solving one equation for one variable and then substituting that expression into the other equation.
• Elimination Method: Adding or subtracting equations to eliminate one variable, making it easier to solve for the other.

For our problem, the substitution method is used due to its simplicity and effectiveness for these particular equations.

Substitution Method

The substitution method is particularly effective for solving systems of equations when one variable is already solved in terms of the other. Here’s how it works in our exercise:

• Identify an equation that gives one variable in terms of the other. In this case, we start with \(x = y + 2\).
• Substitute this expression for \(x\) into the other equation \(2x = 3y - 4\). This results in a single equation with one variable: \(2(y + 2) = 3y - 4\).
• Simplify and solve for \(y\); here, it simplifies to \(y = 8\).
• Substitute the value of \(y\) back into the expression \(x = y + 2\) to find \(x\), which is \(x = 10\).

This method is useful because it breaks a complex system into simpler steps, allowing us to solve for each variable efficiently.