The equation \( y = -5x + 1 \) is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.
1. Identify the Slope and Y-Intercept:
- The slope (\( m \)) of the line is \(-5\).
- The \( y \)-intercept (\( b \)) is \( 1 \).
2. Plotting the Y-Intercept:
- The \( y \)-intercept is the point where the line crosses the \( y \)-axis. For the equation \( y = -5x + 1 \), this point is (0, 1).
3. Using the Slope:
- The slope \(-5\) tells us that for each unit increase in \( x \), \( y \) decreases by 5 units.
- This slope can be represented as a rise over run, which is \(-5/1\). So, starting from the \( y \)-intercept (0, 1), if you move 1 unit to the right (positive direction on the x-axis), you move 5 units down.
4. Sketching the Line:
- Plot the \( y \)-intercept point (0, 1).
- From (0, 1), move 1 unit to the right (to (1, 1)) and 5 units down to (-4).
- Draw a line through these points extending infinitely in both directions.
5. Interpreting the Graph:
- The graph of \( y = -5x + 1 \) is a straight line.
- A line represents all solutions to the equation, meaning every point \((x, y)\) on this line satisfies the equation \( y = -5x + 1 \).
Based on this analysis, the correct option is:
D. a line that shows the set of all solutions to the equation.
