To determine which graph shows the solution to the system of linear equations given by:
- \( y = 4x - 1 \) (Equation 1)
- \( 16x - 4y = 16 \) (Equation 2)
First, we can convert Equation 2 into slope-intercept form (y = mx + b) for easier comparison with Equation 1. Starting from Equation 2:
\[ 16x - 4y = 16 \]
We can solve for \( y \):
\[ -4y = -16x + 16 \]
Dividing everything by -4 gives:
\[ y = 4x - 4 \]
Now we have both equations in slope-intercept form:
- \( y = 4x - 1 \)
- \( y = 4x - 4 \)
Both lines have the same slope of 4, but different y-intercepts (-1 and -4 respectively). Since they have the same slope, they are parallel lines and they will never intersect. This means the system of equations has no solution (they do not meet at any point).
Now, looking at the graph options you provided:
- The first option shows one line that passes through (0, 1) and (1, 2), which corresponds to the equation \( y = 4x - 1 \).
- The second option shows two lines, one that passes through (0, -4), which corresponds to \( y = 4x - 4 \), and the other line through (0, 1) and (1, 2).
- The third option shows a single line that passes through (0, -4) and (1, 0), which does not correspond to either equation.
- The fourth option shows two lines, but the points do not correspond to either equation.
Based on the analysis, the correct graph that represents the two equations given is the second option. However, note that these two lines are parallel and do not intersect, indicating that there is no solution to the system of equations.