Problem 62 - Introductory Algebra for College Students

Question

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{3}-\frac{y}{2}=\frac{2}{3} \\ \frac{2 x}{3}+y=\frac{4}{3} \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2, y = 0\)

Step 1: Transform Fractional Coefficients

It's a good first step to get rid of the fractions by multiplying each equation by numbers that make the denominators cancel out. Multiply the first equation by 6 and the second equation by 3.

Step 2: Apply the Transformation

After multiplication, the equations become: \n\[\begin{align*}2x - 3y = 4 \2x + 3y = 4\end{align*}\]

Step 3: Addition of the two equations

Add the above two equations, we get \(4x = 8\)

Step 4: Solve for x

Next, solve for the variable x by dividing the equation by 4, we get \(x = 2\)

Step 5: Substitution into the second equation

Substitute \(x = 2\) into the second equation \(2x + 3y = 4\) and solve for y.

Step 6: Solve for y

We find that \(y = 0\)

Key Concepts

Algebraic Methods

When tackling systems of equations, algebraic methods provide a structured way to find the values of unknown variables that satisfy all equations simultaneously. These methods are essential because they can be applied to a broad range of problems, from simple linear equations to more complex situations.

Two primary algebraic methods, the elimination method and the substitution method, are commonly used in algebra. Both approaches aim to reduce a system of equations down to one equation with one variable, or to a form that can be readily solved. While these methods may be applied in various contexts, in this exercise, we focus on linear equations.

The process usually begins with transforming the given equations to a more manageable form. This often involves clearing fractions or decimals, combining like terms, and arranging the equations in a specific order. The idea is to simplify the computational steps, reduce potential errors, and make the problem more understandable. Understanding algebraic methods is crucial for students because it cultivates problem-solving skills and logical thinking which are valuable in diverse academic and real-world scenarios.

Elimination Method

The elimination method involves altering the given equations in a system in such a way that adding or subtracting them eliminates one of the variables. This is aimed at reducing a system of multiple equations to a single equation in one variable.

To employ the elimination method effectively, you might need to multiply one or both equations by a number that will make the coefficients of one variable opposites. Once you have opposites, you can add or subtract the equations to eliminate that variable. Continuing with our example, the step-by-step solution demonstrates this by multiplying the equations, which in turn cancels out the variable 'y' when they are added together.

After successfully eliminating one variable, the next step is to solve for the remaining variable. Then, you can substitute the value of this variable back into one of the original equations to solve for the eliminated variable. This dual-step process is a classical move in the elimination method.

Substitution Method

The substitution method is another fundamental technique for solving systems of equations. Instead of eliminating a variable, as seen in the elimination method, this method involves expressing one variable in terms of the other and then substituting this expression into the other equation.

For instance, if one of the equations in the system can be manipulated to isolate 'x' or 'y,' this expression for 'x' or 'y' can then be used in place of the variable in the other equation. This will give us a single equation with a single variable that can be solved. As seen in the step-by-step solution, once 'x' was found to be 2 through elimination, it was substituted back into one of the transformed equations to find 'y'.

This method is particularly useful when one equation is already solved for one variable or can be easily manipulated to do so. The substitution method often provides a pathway to a solution with less chance of arithmetic mistakes and can be more straightforward for visual learners who need to see the progress of one variable being replaced by another.