Sure, let's convert each equation to slope-intercept form and determine how many solutions the system has step by step.
Given equations:
[tex]\[
\begin{array}{r}
-2x + 7y = 1 \\
-4x + 14y = 2
\end{array}
\][/tex]
### Converting the First Equation to Slope-Intercept Form
1. Start with the first equation:
[tex]\[
-2x + 7y = 1
\][/tex]
2. Solve for [tex]\( y \)[/tex] by isolating it on one side of the equation:
[tex]\[
7y = 2x + 1
\][/tex]
3. Divide everything by 7 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
Thus, the first equation in slope-intercept form is:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
### Converting the Second Equation to Slope-Intercept Form
1. Start with the second equation:
[tex]\[
-4x + 14y = 2
\][/tex]
2. Solve for [tex]\( y \)[/tex] by isolating it on one side of the equation:
[tex]\[
14y = 4x + 2
\][/tex]
3. Divide everything by 14 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{4}{14}x + \frac{2}{14}
\][/tex]
4. Simplify the fractions:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
Thus, the second equation in slope-intercept form is:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
### Determine the Number of Solutions
Now that we have both equations in slope-intercept form:
1. First equation:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
2. Second equation:
[tex]\[
y = \frac{2}{7}x + \frac{1}{7}
\][/tex]
We see that both equations are identical. This means they represent the exact same line.
Since the equations represent the same line, the system has infinitely many solutions.
