Question

how many solutions does the system of equations below have?
7x - 6y = 4
12x + 5y = 12
no solution
one solution
infinitely many solutions
submit

Explanation:

Step1: Analyze the slopes of the lines

For a linear equation in the form \(ax + by = c\), the slope \(m\) is given by \(-\frac{a}{b}\).
For the first equation \(7x - 6y = 4\), the slope \(m_1 = -\frac{7}{-6}=\frac{7}{6}\).
For the second equation \(12x + 5y = 12\), the slope \(m_2 = -\frac{12}{5}\).

Step2: Compare the slopes

Since \(m_1=\frac{7}{6}\) and \(m_2 = -\frac{12}{5}\) are not equal, the two lines are not parallel. Also, since the slopes are not negative reciprocals (the product of the slopes \(\frac{7}{6}\times(-\frac{12}{5})=-\frac{14}{5}
eq - 1\)), the lines are not perpendicular but they will intersect at exactly one point.

Answer:

one solution