Question

How many solutions does the equation below have? $$ 3 x-4 y=73 $$ (A) None (B) Exactly 1 (C) Exactly 2 (D) Infinite

Short answer

Infinite solutions exist for this equation.

Step-by-step solution

Analyze the Equation Format

The given equation is a linear Diophantine equation in the form \(3x - 4y = 73\), where coefficients 3 and -4 are integers. We need to determine the number of integer solutions for this equation.

Identify Conditions for Solutions

A linear Diophantine equation \(ax + by = c\) has integer solutions if and only if the greatest common divisor (GCD) of \(a\) and \(b\) divides \(c\). Calculate the GCD of our coefficients, 3 and 4.

Calculate the GCD

Using the Euclidean algorithm, the GCD of 3 and 4 is 1, since they are coprime integers. Since 1 divides any integer, our condition for the equation to have integer solutions is satisfied.

Determine the Number of Solutions

Since our equation is a linear Diophantine equation and the condition is satisfied (GCD divides 73), the equation \(3x - 4y = 73\) will have an infinite number of integer solutions

Key concepts

Integer Solutions in Diophantine Equations

In Diophantine equations, we specifically look for integer solutions where both variables must be whole numbers. These types of equations engage us to find solutions limited to integers due to their specific nature. The most common form of Diophantine equations is the linear form, expressed as \(ax + by = c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are the variables for which we want to find integer solutions.
For the equation \(3x - 4y = 73\), both \(x\) and \(y\) must be integers. We determine the existence and nature of these solutions by examining the coefficients of \(x\) and \(y\).
When dealing with integer solutions:

• Every solution must satisfy the equation completely with whole numbers.
• Specific conditions need to be met, usually involving divisibility and GCD, to ensure solutions exist.

Understanding whether integer solutions exist helps to know how the variables relate under these criteria.

Linear Equations

A linear equation refers to an equation of the first degree, meaning each term is either a constant or the product of a constant and a single variable. Linear equations are fundamental in algebra, commonly represented as \(ax + by = c\). This form can take many shapes, but the key is its straight-line representation on a graph.

• For \(3x - 4y = 73\), the equation is linear as both \(x\) and \(y\) are raised to the power of one.
• Linear equations can have none, exactly one, or infinitely many solutions based on the system they form with other equations.

In the context of a Diophantine equation, linearity implies the simplicity in how \(x\) and \(y\) relate, but it hinges on specific conditions, like the divisibility discussed.
Solutions, whether one or infinitely many, are determined not just by the structure but by the interplay of coefficients of \(x\) and \(y\) with the constant \(c\).

Euclidean Algorithm for GCD

The Euclidean algorithm is a highly efficient method to find the greatest common divisor (GCD) of two integers. The GCD is a crucial part of solving linear Diophantine equations as it informs you whether there will be integer solutions.
To find the GCD of the coefficients in our equation \(3x - 4y = 73\), we apply the Euclidean algorithm: repeatedly subtract the smaller number from the larger until zero is reached. Alternatively, divide the larger number by the smaller number and use the remainder to repeat the steps.

• For 3 and 4, since they are coprime (no common divisor other than 1), the GCD is 1.
• This result is crucial: any integer is divisible by 1, confirming the equation can have solutions.

The GCD helps establish the structure of solutions and confirms whether the equation \(ax + by = c\) can be satisfied by integer values for \(x\) and \(y\). Without the GCD, the understanding of possible solutions would be incomplete.