Problem 189 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. Three times a number plus three times a second number is fifteen. Four times the first plus twice the second number is fourteen. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 2 and 3.

Step 1 - Define Variables

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

Step 2 - Translate Statements into Equations

The first statement can be written as: 3x + 3y = 15. The second statement can be written as: 4x + 2y = 14.

Step 3 - Simplify the Equations

Equation 1: 3x + 3y = 15 can be simplified by dividing all terms by 3: x + y = 5.Equation 2: 4x + 2y = 14 can be simplified by dividing all terms by 2: 2x + y = 7.

Step 4 - Solve the System Using Substitution or Elimination (Substitution Method)

From the simplified Equation 1: x + y = 5 Solve for y: y = 5 - x. Substitute y in the second simplified equation: 2x + (5 - x) = 7.

Step 5 - Simplify and Solve for x

Substitute y in Equation 2: 2x + 5 - x = 7 Combine like terms: x + 5 = 7 Subtract 5 from both sides: x = 2.

Step 6 - Solve for y

Use the value of x in y = 5 - x: y = 5 - 2 y = 3.

Step 7 - Verify the Solution

Substitute x and y back into the original equations to verify: 3(2) + 3(3) = 15 6 + 9 = 15 True 4(2) + 2(3) = 14 8 + 6 = 14 True.

Key Concepts

substitution method

The substitution method is a straightforward way to solve systems of equations. This method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the other equation(s). This allows you to reduce the number of variables and solve step by step.

In our exercise, we started with two equations:
\(3x + 3y = 15\)
and
\(4x + 2y = 14\).
We simplified them to:
\(x + y = 5\)
and
\(2x + y = 7\).
By solving \(x + y = 5\) for \(y\), we get \( y = 5 - x \). This expression for \( y \) can then be substituted in the second equation \(2x + (5 - x) = 7 \).
This step-by-step approach helps to isolate and solve for each variable systematically.

algebraic expressions

Algebraic expressions form the foundation of equations. They consist of variables, constants, and arithmetic operations. Each term in an algebraic expression can represent part of the equation.

In our exercise, we dealt with expressions like \(3x + 3y\) and \(4x + 2y\). These are combinations of coefficients (3, 4, and 2) and variables (\( x \) and \( y \)).
When creating and simplifying algebraic expressions:

• Pay attention to the coefficients
• Combine like terms
• Reduce to simpler forms when possible

For instance,
\ 3x + 3y = 15 \ can be simplified to \ x + y = 5 \ by dividing every term by 3. Simplification makes the problem more manageable.

variables

Variables are symbols that represent unknown values. In equations, they allow us to generalize problems and find solutions for multiple scenarios.

In this exercise, \( x \) and \( y \) are our variables, representing two unknown numbers.
Here’s how you work with variables:

• Define the variables clearly. For instance, let \( x \) be the first number and \( y \) the second number.
• Translate the word problem into mathematical equations using these variables.
• Solve the equations step by step to find the values of the variables.

Identifying variables correctly is crucial because it sets the stage for translating real-world problems into solvable mathematical equations.