Answer :
To determine the correct answer, let's carefully analyze the equation and the graph it produces.
The given equation is [tex]\( y = -2x + 10 \)[/tex]. This is a linear equation, which can be recognized by its form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step-by-Step Solution:
1. Identify the slope and y-intercept:
- The slope ([tex]\( m \)[/tex]) is [tex]\(-2\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( 10 \)[/tex].
2. Understanding the graph of [tex]\( y = -2x + 10 \)[/tex]:
- Since this is a linear equation, its graph will be a straight line.
- The line will cross the y-axis at [tex]\( 10 \)[/tex] (the y-intercept).
3. Nature of the graph:
- For any value of [tex]\( x \)[/tex], we can find a corresponding value of [tex]\( y \)[/tex] using the equation [tex]\( y = -2x + 10 \)[/tex].
- Since we can plot infinitely many points that satisfy this equation, joining these points will form a straight line.
- This line represents all possible solutions to the equation [tex]\( y = -2x + 10 \)[/tex].
### Analysis of the Given Options:
- Option A: "a point that shows the y-intercept."
- This is incorrect because the graph of the equation is not just a single point; it is a line.
- Option B: "a line that shows only one solution to the equation."
- This is incorrect because the line represents an infinite set of solutions, not just one.
- Option C: "a point that shows one solution to the equation."
- This is incorrect for the same reason as Option A; the graph is a line, not a single point.
- Option D: "a line that shows the set of all solutions to the equation."
- This is correct because the line represents all possible (x, y) pairs that satisfy the equation [tex]\( y = -2x + 10 \)[/tex].
### Conclusion:
By understanding the nature of the linear equation and its graph, we can conclude that the correct answer is:
D. a line that shows the set of all solutions to the equation.
The given equation is [tex]\( y = -2x + 10 \)[/tex]. This is a linear equation, which can be recognized by its form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step-by-Step Solution:
1. Identify the slope and y-intercept:
- The slope ([tex]\( m \)[/tex]) is [tex]\(-2\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( 10 \)[/tex].
2. Understanding the graph of [tex]\( y = -2x + 10 \)[/tex]:
- Since this is a linear equation, its graph will be a straight line.
- The line will cross the y-axis at [tex]\( 10 \)[/tex] (the y-intercept).
3. Nature of the graph:
- For any value of [tex]\( x \)[/tex], we can find a corresponding value of [tex]\( y \)[/tex] using the equation [tex]\( y = -2x + 10 \)[/tex].
- Since we can plot infinitely many points that satisfy this equation, joining these points will form a straight line.
- This line represents all possible solutions to the equation [tex]\( y = -2x + 10 \)[/tex].
### Analysis of the Given Options:
- Option A: "a point that shows the y-intercept."
- This is incorrect because the graph of the equation is not just a single point; it is a line.
- Option B: "a line that shows only one solution to the equation."
- This is incorrect because the line represents an infinite set of solutions, not just one.
- Option C: "a point that shows one solution to the equation."
- This is incorrect for the same reason as Option A; the graph is a line, not a single point.
- Option D: "a line that shows the set of all solutions to the equation."
- This is correct because the line represents all possible (x, y) pairs that satisfy the equation [tex]\( y = -2x + 10 \)[/tex].
### Conclusion:
By understanding the nature of the linear equation and its graph, we can conclude that the correct answer is:
D. a line that shows the set of all solutions to the equation.
