Question

how many solutions does the system of equations below have?
9x + 9y = 4
-19x - y = 2
no solution
one solution
infinitely many solutions
submit

Explanation:

Step1: Analyze the slopes of the lines

First, rewrite both equations in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

For the first equation \(9x + 9y=4\):
Subtract \(9x\) from both sides: \(9y=-9x + 4\)
Divide both sides by 9: \(y=-x+\frac{4}{9}\). The slope \(m_1=- 1\).

For the second equation \(-19x - y = 2\):
Add \(19x\) to both sides: \(-y=19x + 2\)
Multiply both sides by - 1: \(y=-19x-2\). The slope \(m_2=-19\).

Step2: Determine the number of solutions

Since the slopes of the two lines (\(m_1=-1\) and \(m_2 = - 19\)) are not equal, the two lines are not parallel and they are not coincident (because coincident lines have the same slope and the same y - intercept). So, the two lines will intersect at exactly one point. A system of linear equations represents two lines, and if the lines intersect at one point, the system has one solution.

Answer:

one solution