Understanding the Given System of Equations

We are given a system of equations with two linear equations:

1. \(3x - 4y = 5\)
2. \(12x - 16y = 20\)

We need to find out how many common solutions these two linear equations have.

Analyzing the Relationship Between the Equations

To determine the number of solutions for a system of linear equations, we can compare the coefficients of the variables and the constant terms. For a general system of two linear equations in two variables:

\(a_1x + b_1y = c_1\)

\(a_2x + b_2y = c_2\)

We look at the ratios of the corresponding coefficients: \(\frac{a_1}{a_2}\), \(\frac{b_1}{b_2}\), and \(\frac{c_1}{c_2}\).

For our given system of equations:

1. \(3x - 4y = 5\) (Here, \(a_1 = 3\), \(b_1 = -4\), \(c_1 = 5\))
2. \(12x - 16y = 20\) (Here, \(a_2 = 12\), \(b_2 = -16\), \(c_2 = 20\))

Calculating the Ratios

Let's calculate the ratios of the coefficients:

• Ratio of x-coefficients: \(\frac{a_1}{a_2} = \frac{3}{12} = \frac{1}{4}\)
• Ratio of y-coefficients: \(\frac{b_1}{b_2} = \frac{-4}{-16} = \frac{1}{4}\)
• Ratio of constant terms: \(\frac{c_1}{c_2} = \frac{5}{20} = \frac{1}{4}\)

We observe that the ratios are equal:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

\(\frac{3}{12} = \frac{-4}{-16} = \frac{5}{20} = \frac{1}{4}\)

Interpreting the Ratios for System of Equations

When the ratios of all corresponding coefficients and constant terms are equal (\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)), it means that the two linear equations are dependent and represent the same line. In such a case, every solution of one equation is also a solution of the other equation.

This type of system of equations is called a consistent system with dependent equations. Geometrically, the lines coincide.

The number of common solutions for coinciding lines is infinite, as every point on the line is a solution to both equations. Infinite solutions are certainly more than two common solutions.

Alternatively, we can notice that the second equation \(12x - 16y = 20\) can be obtained by multiplying the first equation \(3x - 4y = 5\) by 4:

\(4 \times (3x - 4y) = 4 \times 5\)

\(12x - 16y = 20\)

Since one equation is a multiple of the other, they represent the same set of points in the coordinate plane. Therefore, they share infinitely many common solutions.

Conclusion on Common Solutions

Based on the analysis of the ratios and the relationship between the two linear equations, the given system of equations has infinitely many solutions. In the context of the provided options, "more than two common solutions" is the correct description for infinitely many solutions.

Our system of equations matches the third condition, leading to infinitely many common solutions.