Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Use the given statements to write a system of equations. Solve the system by elimination. The sum of twice a number \(r\) and a number \(s\) is 8 . The difference of \(r\) and \(s\) is 7 .

Step-by-Step Solution

Answer

The solution to the system of equations is \(r = 5\) and \(s = -2\).

Step 1: Translate the Problems Statements Into a System of Equations

The first statement can be translated as \(2r + s = 8\). The second statement can be translated as \(r - s = 7\). Thus, the system of equations is: \[\begin{cases} 2r + s = 8 \ r - s = 7 \end{cases}\]

Step 2: Apply Elimination Method

To use elimination, add the two equations of the system together: \((2r + s) + (r - s) = 8 + 7\). This simplifies to \(3r = 15\).

Step 3: Solve for the First Variable

To solve for \(r\), divide both sides of the equation by 3, which gives \(r = 15/3 = 5\).

Step 4: Solve for the Second Variable

Substitute \(r = 5\) into the second equation of the system: \(5 - s = 7\). Solving this equation for \(s\) gives \(s = 5 - 7 = -2\).

Key Concepts

elimination method

The elimination method is a powerful algebraic tool for solving a system of equations. This method involves eliminating one of the variables by adding or subtracting equations in such a way that one variable cancels out, allowing you to solve for the other variable. In our example, we have the system of equations:

• Equation 1: \(2r + s = 8\)
• Equation 2: \(r - s = 7\)

To apply the elimination method, you need to combine these two equations. This can be achieved by adding them together or subtracting one from the other. In our case, we add them:

• \((2r + s) + (r - s) = 8 + 7\)

This gets rid of \(s\) because \(+s - s = 0\), simplifying the equation to \(3r = 15\). With this simpler equation, you can easily solve for \(r\). The elimination method is especially useful when both equations in the system are arranged such that adding or subtracting can cancel out one of the variables.

algebraic translation

Algebraic translation refers to the process of converting word problems or statements into mathematical equations. This is a crucial skill because it allows us to work with numerical and algebraic solutions, rather than relying solely on verbal descriptions. In our provided scenario, let's break down the translation:

• "The sum of twice a number \(r\) and a number \(s\) is 8" translates to the equation \(2r + s = 8\).
• "The difference of \(r\) and \(s\) is 7" translates to the equation \(r - s = 7\).

This process of translation is fundamental as it sets the stage for applying mathematical techniques like the elimination method. It is critical to interpret phrases like "sum", "difference", "product", and "quotient" correctly to ensure accurate mathematical representation. Each part of the statement corresponds to parts of the equations we implement in solving the system.

solving equations

Solving equations is the process of finding the value of the variables in a mathematical equation. With the elimination method, as described above, the process becomes straightforward after setting up the algebraic equations. Let's continue from where the elimination method left off:

• Once you have \(3r = 15\), solve for \(r\) by dividing both sides by 3: \(r = 5\).
• Now, substitute the value of \(r\) back into one of the original equations to find \(s\).

Using the second equation, \(r - s = 7\):

• Substitute \(r = 5\) in to get: \(5 - s = 7\).
• Solve for \(s\) by subtracting 5 from both sides: \(-s = 2\).
• Thus, \(s = -2\).

By solving these equations, you've found that \(r = 5\) and \(s = -2\). This logical sequence allows you to solve any pair of equations systematically. Remember, solving equations is about balancing both sides and isolating the unknown variables.