Answer :
To determine the number of intersections of the graphs of the given equations, let's analyze the system of linear equations.
The equations are:
1. [tex]\( \frac{1}{2} x + 5y = 6 \)[/tex]
2. [tex]\( 3x + 30y = 36 \)[/tex]
We can rewrite these equations in a more standard form without fractions. This often helps make the comparison between the coefficients clearer.
First, let's convert the first equation:
[tex]\[ \frac{1}{2} x + 5y = 6 \][/tex]
Multiply every term by 2 to clear the fraction:
[tex]\[ x + 10y = 12 \][/tex]
Now the first equation is:
[tex]\[ x + 10y = 12 \][/tex]
The second equation is already in a standard form but can also be rewritten to better align with the first equation:
[tex]\[ 3x + 30y = 36 \][/tex]
We can simplify the second equation by dividing every term by 3:
[tex]\[ x + 10y = 12 \][/tex]
So, the simplified system of equations is:
1. [tex]\( x + 10y = 12 \)[/tex]
2. [tex]\( x + 10y = 12 \)[/tex]
Notice that both equations are actually identical. This means that every solution to the first equation is also a solution to the second equation. In terms of their graphical representations, the two lines represented by these equations are the same line.
This implies that there are not a single point of intersection, nor distinct points of intersection, but rather an infinite number of points where these graphs intersect because they are the same line.
Thus, the number of intersection points is:
- Infinitely many
So, the answer is:
infinitely many
The equations are:
1. [tex]\( \frac{1}{2} x + 5y = 6 \)[/tex]
2. [tex]\( 3x + 30y = 36 \)[/tex]
We can rewrite these equations in a more standard form without fractions. This often helps make the comparison between the coefficients clearer.
First, let's convert the first equation:
[tex]\[ \frac{1}{2} x + 5y = 6 \][/tex]
Multiply every term by 2 to clear the fraction:
[tex]\[ x + 10y = 12 \][/tex]
Now the first equation is:
[tex]\[ x + 10y = 12 \][/tex]
The second equation is already in a standard form but can also be rewritten to better align with the first equation:
[tex]\[ 3x + 30y = 36 \][/tex]
We can simplify the second equation by dividing every term by 3:
[tex]\[ x + 10y = 12 \][/tex]
So, the simplified system of equations is:
1. [tex]\( x + 10y = 12 \)[/tex]
2. [tex]\( x + 10y = 12 \)[/tex]
Notice that both equations are actually identical. This means that every solution to the first equation is also a solution to the second equation. In terms of their graphical representations, the two lines represented by these equations are the same line.
This implies that there are not a single point of intersection, nor distinct points of intersection, but rather an infinite number of points where these graphs intersect because they are the same line.
Thus, the number of intersection points is:
- Infinitely many
So, the answer is:
infinitely many
