Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{aligned} \frac{x+3}{4}+\frac{y-1}{3} &=1 \\ x-y &=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system is (19/7, -2/7). The system is consistent.

Step 1: Rewrite Equations

The first step is to rewrite the equations in a more standard form. The first equation can be simplified by multiplying every term by 12 (which is the least common multiple of 4 and 3). Call this equation (1). Thus, the system of equations to solve becomes: \[3x + 9 + 4y - 4 = 12\] and \[x - y = 3\]

Step 2: Simplify Equation

Now simplify equation (1) further to: \[3x + 4y = 7\] and rewrite equation (2) as \[x = y + 3\] for later substitution.

Step 3: Substitute into Equation (1)

Substitute equation (2) into equation (1) to isolate one variable: \[3(y + 3) + 4y = 7\]

Step 4: Simplify to Find y

Simplify to find the value of 'y'. This gives \[3y + 9 + 4y = 7\] or \[7y = -2\] and thus, \[y = -2/7\].

Step 5: Substitute y into Equation (2)

Now that the value of y is known, substitute it into equation (2) to find the value of 'x'. This gives: \[x = -(2/7) + 3\]

Step 6: Simplify to Find x

Simplify further to find the value of 'x', which is: \[x = 19/7\] and therefore the solution pairs is \[(19/7, -2/7)\]

Step 7: Define if Consistent or Inconsistent

Because a solution that fits both equations has been found, this system is consistent.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. It involves combining the equations in a way that eliminates one of the variables, making it easier to solve for the remaining variable.
Here's how it typically works:

• You start by aligning the equations in standard form, with like terms in columns. This was done in the exercise by rewriting and simplifying the original equations.
• The goal is to manipulate one or both equations so that a variable can be eliminated. This is often achieved by adding or subtracting the equations after multiplying them by suitable coefficients.
• Once a variable is eliminated, the system reduces to a single equation, allowing you to solve for the other variable directly.

In the given exercise, the elimination method was made simpler through substitution after aligning the equations, to isolate and solve for the variables. This shows the adaptability of the elimination method, where it can blend with substitution to efficiently solve the system.

Consistent System

A system of equations is consistent when there is at least one set of values for the variables that satisfies all the equations in the system. This means that the equations intersect at a point, providing a common solution.
In contrast, an inconsistent system has no solutions, which generally indicates that the equations represent parallel lines that never intersect.
For the given exercise:

• By solving the system using the elimination method, a specific solution was found: \[(x = 19/7, y = -2/7)\]
• This solution satisfies both equations upon substitution, confirming the consistency of the system.
• The identification of a consistent system is crucial in determining that the problem has a meaningful answer and that the equations are properly representative of the scenario being modeled.

Therefore, recognizing a consistent system is vital for validating the results obtained from the chosen solution method.

Substitution Method

The substitution method is another essential strategy for solving systems of equations. This approach involves solving one equation for one variable and then substituting that expression into another equation.
The process is straightforward:

• The first step is to isolate one variable in one of the equations—just like it was done with the equation \( x = y + 3 \) in the exercise.
• Next, substitute the expression for this isolated variable into the other equation. This reduces the system to a single equation with one variable.
• After substituting, solve the resulting equation to find the value of the isolated variable.
• Finally, substitute back to find the value of the other variable, ensuring both solutions satisfy the original equations.

In the exercise provided, the substitution method was used effectively after simplifying the equations. This helped to solve for 'y' and later 'x', demonstrating its efficacy in breaking down complex systems into manageable parts.