Problem 187 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. Twice a number plus three times a second number is twentytwo. Three times the first number plus four times the second is thirty-one. Find the numbers.

Step-by-Step Solution

Answer

The first number is 5 and the second number is 4.

Step 1: Define Variables

Let x be the first number and y be the second number.

Step 2: Translate to Equations

From the problem statement, we can write the following equations: Equation 1: \( 2x + 3y = 22 \) Equation 2: \( 3x + 4y = 31 \)

Step 3: Use the Method of Elimination

First, multiply Equation 1 by 3 and Equation 2 by 2 to eliminate x: \( 3(2x + 3y) = 3(22) \) \( 2(3x + 4y) = 2(31) \) This gives us: \( 6x + 9y = 66 \) \( 6x + 8y = 62 \)

Step 4: Subtract and Solve for y

Subtract the second new equation from the first: \( (6x + 9y) - (6x + 8y) = 66 - 62 \) This simplifies to: \( y = 4 \)

Step 5: Substitute y Back into One of the Original Equations

Substitute \( y = 4 \) into Equation 1: \( 2x + 3(4) = 22 \) This simplifies to: \( 2x + 12 = 22 \)

Step 6: Solve for x

Subtract 12 from both sides: \( 2x = 10 \) Divide both sides by 2: \( x = 5 \)

Step 7: Write the Final Answer

The first number is \( x = 5 \) and the second number is \( y = 4 \).

Key Concepts

Elimination Method

The elimination method is a powerful way to solve a system of equations. It focuses on eliminating one variable by adding or subtracting equations. In our exercise, we have two equations:
Equation 1: \( 2x + 3y = 22 \)
Equation 2: \( 3x + 4y = 31 \)
To eliminate a variable, we need to manipulate the equations. We'll multiply each to align the coefficients of one variable.
We chose to eliminate \( x \) by multiplying Equation 1 by 3 and Equation 2 by 2:

• \( 3(2x + 3y) = 3(22) \)
• \( 2(3x + 4y) = 2(31) \)

This transforms the equations to:
\( 6x + 9y = 66 \)
\( 6x + 8y = 62 \)
By subtracting the second from the first, we eliminate \( x \) and solve for \( y \):
\( (6x + 9y) - (6x + 8y) = 66 - 62 \)
\( y = 4 \)

Algebraic Equations

Algebraic equations represent relationships between variables. They involve constants and coefficients. Our two equations:
Equation 1: \( 2x + 3y = 22 \)
Equation 2: \( 3x + 4y = 31 \)
show the relationship between the two numbers we need to find. The coefficients are the numbers multiplying the variables. In these equations:

• 2 and 3 are coefficients in Equation 1
• 3 and 4 are coefficients in Equation 2

The constants on the right side (22 and 31) are the total sums. By understanding these parts, we get a clear view of how the variables relate to each other.

Variable Substitution

Variable substitution is a method where we replace one variable with an expression involving the other variable. After using the elimination method, we found \( y = 4 \). Next, we substitute \( y = 4 \) back into one of the original equations to find \( x \).
We chose Equation 1:
\( 2x + 3(4) = 22 \)
This simplifies to:

• \( 2x + 12 = 22 \)

By isolating \( x \), we subtract 12 from both sides:
\( 2x = 10 \)
Then, we divide both sides by 2 to solve for \( x \):
\( x = 5 \).
This method helps us find the value of one variable using the value of the other.

Solving Equations

Solving equations is about finding the value of unknown variables that satisfy the equation. In our exercise, we solved the system using the following process:

• Translate word problem to equations
• Use elimination to eliminate one variable
• Solve the remaining equation for one variable
• Substitute back to find the other variable

Our final answers were:
First number: \( x = 5 \)
Second number: \( y = 4 \)
By practicing these steps frequently, you can become more efficient and accurate in solving algebraic equations.