Problem 28 - Thinking Mathematically

Question

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x+2 y=14 \\ 3 x-2 y=10\end{array}\right.\)

Step-by-Step Solution

Answer

Solving the system of equations gives the solution as \(x = 4\) and \(y = 1\).

Step 1: Set up the equations

The system of equations is given as: \(\left\{\begin{array}{l}3 x+2 y=14 \ 3 x-2 y=10\end{array}\right.\). This system can be written one below the other to facilitate the process of addition.

Step 2: Add the equations

Add the two equations to eliminate the y variable. Expression: \(3x + 2y + 3x - 2y = 14 + 10\). This simplifies to: \(6x = 24\).

Step 3: Solve for x

Divide both sides of the equation \(6x = 24\) by 6 to find the value of x, which comes out to be \(\frac{24}{6} = 4\).

Step 4: Substitute x into one of the equations

Having found the value of x to be 4, it is substituted into the first equation (3x + 2y = 14) to find the value of y, giving \(3*4 + 2y = 14\). This simplifies to \(12 + 2y = 14\).

Step 5: Solve for y

Subtract 12 from both sides of \(12 + 2y = 14\) you get \(2y = 2\). Then divide by 2 to find the value of y, which yields \(y = \frac{2}{2}=1\).

Step 6: Check the solution

Substitute x = 4 and y = 1 into both equations to verify if they satisfy both the equations.

Key Concepts

Addition Method

The Addition Method is a powerful technique used to solve systems of linear equations. The main goal of this method is to eliminate one of the variables by adding two equations from the system together. This method is particularly useful when the coefficients of one of the variables are opposites or can be easily manipulated to become opposites.

By adding the equations, we can cancel out one of the variables, simplifying the system and allowing us to solve for the remaining variable easily. For example, in the given exercise, the system of equations is:

• Equation 1: \(3x + 2y = 14\)
• Equation 2: \(3x - 2y = 10\)

If we add these equations as shown:\[(3x + 2y) + (3x - 2y) = 14 + 10\]The terms involving \(y\) cancel each other out, leaving us with a new equation \(6x = 24\). This greatly simplifies solving for \(x\). This technique is sometimes referred to as the elimination method because it eliminates one variable.

Substitution

Substitution is a method used to solve systems of equations by expressing one variable in terms of another, then substituting this expression into the remaining equation. This technique often follows the elimination of a variable using the Addition Method.

In our exercise, after finding that \(x = 4\) using the Addition Method, we employ substitution to find \(y\). We substitute \(x\) into either of the original equations. Choosing the first equation:

• \(3x + 2y = 14\)

Replace \(x\) with 4:\[3(4) + 2y = 14\]This simplifies to \(12 + 2y = 14\). By isolating \(y\), we simplify and solve the equation: subtract 12 from both sides and divide by 2, resulting in \(y = 1\). Substitution allows us to find the value of one variable based on the known value of another.

Checking Solutions

Checking solutions is a crucial step in solving systems of equations to ensure accuracy. After solving for both \(x\) and \(y\), it's important to substitute these values back into the original equations to verify that they satisfy both equations.

For our system:

• Equation 1: \(3x + 2y = 14\)
• Equation 2: \(3x - 2y = 10\)

Substitute \(x = 4\) and \(y = 1\) into both equations:

• For Equation 1: \(3(4) + 2(1) = 12 + 2 = 14\) (True)
• For Equation 2: \(3(4) - 2(1) = 12 - 2 = 10\) (True)

Since both equations are satisfied with these values, we can be confident that \((x, y) = (4, 1)\) is indeed the correct solution. Checking solutions not only confirms the correctness but also helps identify any errors made during calculations.

Linear Equations

Linear equations, fundamental components of algebra, represent equations of the first degree. These equations graph as straight lines on a coordinate plane and typically take the form \(ax + by = c\). In our exercise, we deal with two such linear equations:

• \(3x + 2y = 14\)
• \(3x - 2y = 10\)

Each equation describes a line, and the solution to the system is the point where these two lines intersect. This point is represented by specific \(x\) and \(y\) values that satisfy both equations simultaneously. Understanding the nature of linear equations helps to visualize and predict the outcomes when solving systems, as well as recognizing the geometric interpretation of solutions.