Problem 26 - College Algebra Essentials

Question

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x-7 y=13 \\ 6 x+5 y=7 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2, y= -1\).

Step 1: Match the Coefficients

The goal is to make the coefficients of one of the variables identical in both equations so that one can be eliminated through subtraction. To achieve this, multiply the first equation with the number 2, which leads to a new system: \[\left\{\begin{array}{l} 6x-14y=26 \ 6x+5y=7 \end{array}\right. \]

Step 2: Subtract the Equations

Now, subtract the second equation from the first equation to eliminate the variable \(x\):\[(6x -14y) - (6x +5y) = 26 -7\]which simplifies to:\[-19y = 19\]

Step 3: Solve for y

Solve for \(y\) by dividing each side of the equation with -19. It leads to \(y = -1\)

Step 4: Substitute y into one of the Original Equations

Once you solve for one variable, substitute it back into one of the original equations to find the value for \(x\). Substituting \(y = -1\) into the first equation (3x -7y =13), we get:\[3x +7 =13\]

Step 5: Solve for x

Now solve for \(x\) to get \(x = 2\).

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a powerful technique for solving systems of equations. The basic idea is to combine the two equations into one. We aim to eliminate one of the variables by adding or subtracting the equations after aligning coefficients.

Here's how it works in this problem:

• First, we ensure one variable has the same coefficient in both equations. In our example, we multiplied the entire first equation by 2. This made the coefficients of the variable \(x\) the same in both equations.
• After aligning the coefficients, one of the variables can be eliminated. For example, subtracting the second equation from the first eliminated \(x\).

This simplifies the process and lets us solve for the remaining variable. After applying the addition method, students can find the value of one variable, which is an essential step towards solving the entire system.

Substitution Method

The substitution method is another common way to solve systems of equations. Unlike the addition method, this approach involves solving one of the equations for a particular variable and substituting that expression into the other equation. Here’s how this method can complement the process:

• First, solve one of the original equations for one of the variables. In our context, once we found \(y = -1\), this became our stepping stone for substitution.
• Substitute this value back into the other equation. This transforms the system into a single-variable equation. It simplifies the complexity and puts us closer to finding the complete solution by finding the second variable's value.

By using substitution, students are offered an alternative and often more straightforward path to solve the remaining equation once one variable is determined.

Algebraic Solution

Solving a system of equations using algebra involves careful manipulation and logical progression through each step. It's like a puzzle where each piece moves you closer to the complete picture.

• Begin by breaking down the system to the simplest forms possible, either by addition, subtraction, or substitution, which makes the system more manageable.
• Apply algebraic operations like multiplication, division, addition, or subtraction to isolate and solve for variables step-by-step.
• Always check your solution by substituting the values of \(x\) and \(y\) back into the original equations to ensure they satisfy both equations.

This structured algebraic approach ensures accuracy and understanding, reinforcing the core concepts of systems of equations and providing a clear path to the solution. Solving with confidence involves following these strategic steps with patience and precision.