To determine what the graph of the equation [tex]\( y = -4x + 7 \)[/tex] represents, let's analyze the form of the equation and the nature of its graph.
1. Identify the Type of Equation:
The given equation [tex]\( y = -4x + 7 \)[/tex] is in the slope-intercept form of a linear equation, which is generally written as [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line
- [tex]\( b \)[/tex] is the y-intercept, which is the point where the line crosses the y-axis.
2. Understand the Components:
- Here, [tex]\( m = -4 \)[/tex], indicating the slope of the line.
- [tex]\( b = 7 \)[/tex], indicating the y-intercept of the line. So, the line crosses the y-axis at the point (0, 7).
3. Graphical Representation:
- A linear equation like [tex]\( y = -4x + 7 \)[/tex] represents a straight line on a two-dimensional Cartesian coordinate system.
- This line is infinite in length and extends in both directions without end.
- Every point [tex]\((x, y)\)[/tex] on the line is a solution to the equation [tex]\( y = -4x + 7 \)[/tex].
4. Interpretation of the Graph:
- Since a line contains an infinite number of points, each of which is a solution to the equation, the graph is not simply a point.
- Instead, the graph represents the entire collection of solutions that satisfy the given equation.
5. Conclusions:
- Option A states "a point that shows the [tex]\( y \)[/tex]-intercept." This is incorrect because the graph is not just a point; it includes all points that satisfy the equation.
- Option B states "a line that shows only one solution to the equation." This is incorrect because a line represents an infinite set of solutions, not just one.
- Option C states "a line that shows the set of all solutions to the equation." This is the correct description as the line represents all the pairs [tex]\((x, y)\)[/tex] that satisfy the equation [tex]\( y = -4x + 7 \)[/tex].
- Option D states "a point that shows one solution to the equation." This is incorrect because, once again, the graph is not just a single point but a line.
Therefore, the correct answer is:
C. a line that shows the set of all solutions to the equation.
