Question
By inspecting the equations, what can you determine about the solution(s) of this system?
12y=9x+33
20y=15x+55
12y=9x+33
20y=15x+55
Answers
GPT-4o mini
To analyze the system of equations and determine the solutions, let's first rewrite the equations in a standard form.
1. The first equation is:
\[
12y = 9x + 33
\]
Rearranging this to the standard form \(Ax + By = C\), we get:
\[
9x - 12y + 33 = 0 \quad \text{(or } 9x - 12y = -33\text{)}
\]
2. The second equation is:
\[
20y = 15x + 55
\]
Rearranging this to standard form similarly, we get:
\[
15x - 20y + 55 = 0 \quad \text{(or } 15x - 20y = -55\text{)}
\]
Next, let's analyze the coefficients of \(x\) and \(y\) in both equations to check if they are proportional.
- From the first equation \(9x - 12y = -33\):
- Coefficients: \(A_1 = 9\), \(B_1 = -12\)
- From the second equation \(15x - 20y = -55\):
- Coefficients: \(A_2 = 15\), \(B_2 = -20\)
Now, let’s check if the ratios of the coefficients are equal:
\[
\frac{A_1}{A_2} = \frac{9}{15} = \frac{3}{5}
\]
\[
\frac{B_1}{B_2} = \frac{-12}{-20} = \frac{12}{20} = \frac{3}{5}
\]
The coefficients \(A_1/A_2\) and \(B_1/B_2\) are equal, which suggests the equations are proportional. We should also check the constant terms:
\[
\frac{-33}{-55} = \frac{33}{55} = \frac{3}{5}
\]
Since the ratios of all the coefficients (including the constant terms) are the same, this indicates that the two lines represented by the equations are actually the same line.
Thus, the system of equations has infinitely many solutions because the equations are dependent. Each equation is a different representation of the same linear equation.
1. The first equation is:
\[
12y = 9x + 33
\]
Rearranging this to the standard form \(Ax + By = C\), we get:
\[
9x - 12y + 33 = 0 \quad \text{(or } 9x - 12y = -33\text{)}
\]
2. The second equation is:
\[
20y = 15x + 55
\]
Rearranging this to standard form similarly, we get:
\[
15x - 20y + 55 = 0 \quad \text{(or } 15x - 20y = -55\text{)}
\]
Next, let's analyze the coefficients of \(x\) and \(y\) in both equations to check if they are proportional.
- From the first equation \(9x - 12y = -33\):
- Coefficients: \(A_1 = 9\), \(B_1 = -12\)
- From the second equation \(15x - 20y = -55\):
- Coefficients: \(A_2 = 15\), \(B_2 = -20\)
Now, let’s check if the ratios of the coefficients are equal:
\[
\frac{A_1}{A_2} = \frac{9}{15} = \frac{3}{5}
\]
\[
\frac{B_1}{B_2} = \frac{-12}{-20} = \frac{12}{20} = \frac{3}{5}
\]
The coefficients \(A_1/A_2\) and \(B_1/B_2\) are equal, which suggests the equations are proportional. We should also check the constant terms:
\[
\frac{-33}{-55} = \frac{33}{55} = \frac{3}{5}
\]
Since the ratios of all the coefficients (including the constant terms) are the same, this indicates that the two lines represented by the equations are actually the same line.
Thus, the system of equations has infinitely many solutions because the equations are dependent. Each equation is a different representation of the same linear equation.