Question

1. by inspecting the graph, what can you determine about the solution(s) of this system?

20y = 15x + 55
12y = 9x + 33

Explanation:

Step1: Analyze the equations' slopes

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

For the equation \(20y=15x + 55\), divide both sides by 20:
\(y=\frac{15}{20}x+\frac{55}{20}=\frac{3}{4}x+\frac{11}{4}\)
The slope \(m_1=\frac{3}{4}\)

For the equation \(12y = 9x+33\), divide both sides by 12:
\(y=\frac{9}{12}x+\frac{33}{12}=\frac{3}{4}x+\frac{11}{4}\)
The slope \(m_2 = \frac{3}{4}\)

Step2: Analyze the y - intercepts

From the slope - intercept form of the first equation \(y=\frac{3}{4}x+\frac{11}{4}\), the y - intercept \(b_1=\frac{11}{4} = 2.75\)
From the slope - intercept form of the second equation \(y=\frac{3}{4}x+\frac{11}{4}\), the y - intercept \(b_2=\frac{11}{4}=2.75\)

Since the two lines have the same slope (\(m_1 = m_2=\frac{3}{4}\)) and the same y - intercept (\(b_1=b_2=\frac{11}{4}\)), the two lines are coincident (they lie on top of each other).

Step3: Determine the number of solutions

When two lines are coincident, every point on one line is also on the other line. So, the system of equations has infinitely many solutions.

Answer:

The system of equations \(20y = 15x + 55\) and \(12y=9x + 33\) has infinitely many solutions because the two lines represented by the equations are coincident (they have the same slope and the same y - intercept).