Problem 33 - Algebra Form and Function

Question

Without solving the equations, decide how many solutions the system has. $$ \left\\{\begin{array}{l} 4 x-3=y \\ \frac{4}{y}-\frac{1}{x}=\frac{3}{x y} \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The given system of equations has infinitely many solutions.

Step 1: Analyze the first equation

Given the first equation, $$ 4x-3=y $$ We can rewrite it as, $$ y=4x-3 $$ This is a linear equation representing a straight line in the \(xy\)-plane.

Step 2: Analyze the second equation

The second equation is given as: $$ \frac{4}{y}-\frac{1}{x}=\frac{3}{xy} $$ Let's manipulate this equation step by step to find a relationship between \(y\) and \(x\). Step 2.1: Clear the fractions by multiplying through by \(xy\): $$ 4x - y = 3 $$ Step 2.2: Rewrite the equation in terms of \(y\): $$ y = 4x-3 $$

Step 3: Conclusion

Comparing the two equations, we can see that they are identical: $$ y = 4x - 3 $$ As the two equations represent the same line, the system is dependent, meaning there are infinitely many solutions. Each point on the line represented by \(y=4x-3\) is a solution to the system. Therefore, the system has infinitely many solutions.

Key Concepts

Linear Equations

Linear equations are foundational in algebra and represent relationships where the variables appear to the power of one. The general form of a linear equation in two variables is \(ax + by = c\). When graphing, these equations always produce a straight line. Understanding the characteristics of linear equations is key to analyzing systems of equations.

When you rewrite equations to align them with other linear equations, you might notice common features such as a consistent rise over run, otherwise known as the slope. When you write an equation like \(y = mx + b\), \(m\) represents the slope and \(b\) is the y-intercept.

• Linear equations are characterized by constant rates.
• Their graphical representation is a straight line.
• Simplifying equations makes it easier to compare them.

These characteristics make linear equations predictable and straightforward to analyze, particularly in systems where they can be solved visually and algebraically.

Equivalent Equations

Equivalent equations are different representations that have the same solutions. For example, \(4x - 3 = y\) and \(y = 4x - 3\) are equivalent since they describe the same relationship. They may look different, but they visually represent the same line when graphed.

To determine if two equations are equivalent, you can manipulate one equation to see if it can be rearranged into the form of the other. This involves operations such as addition, subtraction, and multiplication that do not change the solution set.

• Rearranging equations by similar operations guides you to equivalence.
• Graphical interpretation often reveals if equations are equivalent.
• Equivalent equations will always intersect at identical points on a graph.

Understanding equivalent equations is crucial in systems of equations, as it helps you identify when different forms describe the same relationship.

Infinite Solutions

A system of equations can have infinitely many solutions when the equations describe the same line. As shown in the given exercise, both equations are equivalent, leading to the conclusion of infinite solutions since they represent the same graph in the coordinate plane.

Infinite solutions indicate that every point on the line satisfies both equations in the system. This does not mean that the solution set is infinite in scope, but rather that there are numerous precise solutions that fit both equations perfectly.

• Infinite solutions imply that one equation is a multiple of another.
• The concept signals a dependent system of equations.
• Every pair \((x, y)\) on the line is a valid solution.

Recognizing infinite solutions in systems of equations is vital, as it affects the approach you take to solving these problems and clarifies that the equations involved are interdependent.