To determine the correct answer to the question about the graph of the equation [tex]\(y = -3x + 4\)[/tex], we need to analyze each given option carefully.
The equation [tex]\(y = -3x + 4\)[/tex] is a linear equation in slope-intercept form, where [tex]\(y\)[/tex] is the dependent variable, [tex]\(x\)[/tex] is the independent variable, the coefficient of [tex]\(x\)[/tex] is the slope of the line, and the constant term (4) is the y-intercept.
1. Option A: a point that shows the [tex]\(y\)[/tex]-intercept.
- The [tex]\(y\)[/tex]-intercept of a linear equation is where the line crosses the [tex]\(y\)[/tex]-axis. While [tex]\(y = 4\)[/tex] is indeed the [tex]\(y\)[/tex]-intercept of [tex]\( y = -3x + 4 \)[/tex], this does not describe the entire graph. It only identifies one specific point on the graph, which is [tex]\((0, 4)\)[/tex].
2. Option B: a line that shows only one solution to the equation.
- A line in a graph represents not just one point, but an infinite number of points that satisfy the equation. Hence, this option is incorrect, as a line represents more than one solution.
3. Option C: a line that shows the set of all solutions to the equation.
- This is accurate because the graph of a linear equation is a straight line, and every point on this line represents a solution [tex]\((x, y)\)[/tex] that satisfies the equation [tex]\(y = -3x + 4\)[/tex]. Therefore, this line represents the set of all solutions.
4. Option D: a point that shows one solution to the equation.
- This option is incorrect because, similar to option A, it refers to a single point rather than the entire set of points forming the line.
Given these explanations, the correct answer is option C.
In conclusion, the graph of [tex]\(y = -3x + 4\)[/tex] is:
C: a line that shows the set of all solutions to the equation.
