Let's analyze each of the given equations to determine which one represents the line with a slope of 2 and a y-intercept of 4.
### Equation 1: [tex]\( y = 2x + 4 \)[/tex]
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In this equation, [tex]\( y = 2x + 4 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is 2.
- The y-intercept ([tex]\( b \)[/tex]) is 4.
- This equation matches the given criteria (a slope of 2 and a y-intercept of 4).
### Equation 2: [tex]\( y = \frac{1}{2}x + 4 \)[/tex]
- In this equation, [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\(\frac{1}{2}\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is 4.
- This equation has the correct y-intercept but the incorrect slope. The slope should be 2, not [tex]\(\frac{1}{2}\)[/tex].
### Equation 3: [tex]\( y = \frac{1}{2}x - 4 \)[/tex]
- In this equation, [tex]\( y = \frac{1}{2}x - 4 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\(\frac{1}{2}\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is -4.
- This equation has neither the correct slope nor the correct y-intercept.
### Equation 4: [tex]\( y = 2x - 4 \)[/tex]
- In this equation, [tex]\( y = 2x - 4 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is 2.
- The y-intercept ([tex]\( b \)[/tex]) is -4.
- This equation has the correct slope but the incorrect y-intercept. The y-intercept should be 4, not -4.
After evaluating all the equations, the one that correctly represents a line with a slope of 2 and a y-intercept of 4 is:
[tex]\[ y = 2x + 4 \][/tex]
Therefore, the correct equation is the first one, and the index of this equation in the list is [tex]\( 0 \)[/tex].
