To determine the correct answer to the question about the graph of the equation [tex]\( y = -3x + 4 \)[/tex], let's break down the components of the equation and understand what they represent.
1. Understanding the Equation:
- The given equation is [tex]\( y = -3x + 4 \)[/tex].
- This is a linear equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Interpreting the Equation:
- The slope [tex]\( m \)[/tex] is -3, which means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
- The y-intercept [tex]\( b \)[/tex] is 4, which means the line crosses the y-axis at the point (0, 4).
3. Graphing the Equation:
- Plot the y-intercept (0, 4) on the graph.
- Use the slope to plot another point. From (0, 4), move 1 unit to the right (positive direction on the x-axis) and 3 units down (negative direction on the y-axis) to get the point (1, 1).
- Draw a straight line through these points, which extends infinitely in both directions.
4. Nature of the Graph:
- The graph of [tex]\( y = -3x + 4 \)[/tex] is a straight line.
- This line represents all the (x, y) pairs that satisfy the equation. Every point on this line is a solution to the equation.
Given the characteristics of the equation and its graph:
- Option A, "a line that shows only one solution to the equation", is incorrect because a line represents infinitely many solutions.
- Option B, "a point that shows one solution to the equation", is incorrect because the graph is a line, not a point.
- Option C, "a point that shows the y-intercept", is partially correct about the y-intercept but does not represent the entire graph.
- Option D, "a line that shows the set of all solutions to the equation", is correct because the graph of a linear equation is a line that includes all the solutions to the equation.
Therefore, the correct answer is:
D. a line that shows the set of all solutions to the equation.
