First, let's analyze the given equation [tex]\( y = -4x + 7 \)[/tex].
1. Form of the Equation: This equation is in the form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. In this case, the slope ([tex]\( m \)[/tex]) is [tex]\(-4\)[/tex], and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(7\)[/tex].
2. Understanding Linear Equations: A linear equation represents a straight line when graphed on the coordinate plane. This line includes all points [tex]\((x, y)\)[/tex] that satisfy the equation. Hence, each point on the line is a solution to the equation.
3. Graphical Representation: Let's consider what the graph tells us:
- The y-intercept ([tex]\( b = 7 \)[/tex]) means that when [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] will be [tex]\( 7 \)[/tex]. So, the line will cross the y-axis at [tex]\( (0, 7) \)[/tex].
- The slope ([tex]\( m = -4 \)[/tex]) indicates that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 4 units.
4. Choices Explanation:
- Choice A: "a point that shows the y-intercept" focuses on just one specific point, which is incomplete for a linear equation.
- Choice B: "a line that shows the set of all solutions to the equation" correctly describes that every point on the line is a solution to the equation, representing a continuous set of solutions.
- Choice C: "a point that shows one solution to the equation" is incorrect because a linear equation in two variables represents all possible solutions along the line, not just one.
- Choice D: "a line that shows only one solution to the equation" is incorrect because a line represents an infinite number of solutions, not just one.
5. Conclusion: The graph of the equation [tex]\( y = -4x + 7 \)[/tex] is a straight line that represents the set of all solutions to the equation.
Hence, the correct answer is:
B. a line that shows the set of all solutions to the equation.
