Sure, let's convert each equation into slope-intercept form, which is written as [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Step 1: Convert the first equation to slope-intercept form.
The first equation is:
[tex]\[ -2x + 7y = 1 \][/tex]
To convert this to slope-intercept form, we need to solve for [tex]\( y \)[/tex].
1. Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[ 7y = 2x + 1 \][/tex]
2. Divide every term by 7:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
So, the first equation in slope-intercept form is:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
Step 2: Convert the second equation to slope-intercept form.
The second equation is:
[tex]\[ -4x + 14y = 2 \][/tex]
To convert this to slope-intercept form, we need to solve for [tex]\( y \)[/tex].
1. Add [tex]\( 4x \)[/tex] to both sides:
[tex]\[ 14y = 4x + 2 \][/tex]
2. Divide every term by 14:
[tex]\[ y = \frac{4}{14}x + \frac{2}{14} \][/tex]
3. Simplify the fractions:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
So, the second equation in slope-intercept form is:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
Step 3: Determine the number of solutions without solving the system.
We have the following two equations in slope-intercept form:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
Since both equations simplify to the same line, the system of equations represents the same line. Therefore, they have:
[tex]\[ \text{Infinite solutions} \][/tex]
In conclusion, the first equation in slope-intercept form is:
[tex]\[ y = \frac{2}{7}x + \frac{1}{7} \][/tex]
The system has:
[tex]\[ \text{Infinite solutions}. \][/tex]
