Answer :
To determine what the graph of the equation \(y = -3x + 4\) represents, let's analyze the equation step by step.
1. Identify the type of equation: The equation \(y = -3x + 4\) is in the form of \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This form is known as the slope-intercept form of a linear equation.
2. Slope and Y-intercept:
- Slope (\(m\)): The slope of the line is \(-3\), which indicates that for every unit increase in \(x\), \(y\) decreases by 3 units.
- Y-intercept (\(b\)): The y-intercept is \(4\), which means the line crosses the y-axis at the point \((0, 4)\).
3. Graphing the line:
- Start by plotting the y-intercept at \((0, 4)\).
- Use the slope to find another point on the line. For instance, if you move 1 unit to the right (increase \(x\) by 1), you move 3 units down (since the slope is \(-3\)). This takes you to the point \((1, 1)\).
- Connect these points with a straight line.
4. Nature of the graph:
- A linear equation in the form \(y = mx + b\) represents a straight line on the Cartesian plane.
- This line extends infinitely in both directions and contains an infinite number of points.
- Each point \((x, y)\) on this line is a solution to the equation \(y = -3x + 4\).
5. Matching to options:
- Option A states "a line that shows the set of all solutions to the equation". This correctly describes the graph of a linear equation.
- Option B states "a point that shows one solution to the equation", which is incorrect because the graph represents more than one solution.
- Option C states "a point that shows the \(y\)-intercept", which is misleading because while the y-intercept is a point on the graph, the entire graph is a line, not just a point.
- Option D states "a line that shows only one solution to the equation", which is incorrect because a line contains infinitely many points, each representing a solution.
Therefore, based on the analysis, the correct answer is [tex]\( \boxed{\text{A}} \)[/tex].
1. Identify the type of equation: The equation \(y = -3x + 4\) is in the form of \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This form is known as the slope-intercept form of a linear equation.
2. Slope and Y-intercept:
- Slope (\(m\)): The slope of the line is \(-3\), which indicates that for every unit increase in \(x\), \(y\) decreases by 3 units.
- Y-intercept (\(b\)): The y-intercept is \(4\), which means the line crosses the y-axis at the point \((0, 4)\).
3. Graphing the line:
- Start by plotting the y-intercept at \((0, 4)\).
- Use the slope to find another point on the line. For instance, if you move 1 unit to the right (increase \(x\) by 1), you move 3 units down (since the slope is \(-3\)). This takes you to the point \((1, 1)\).
- Connect these points with a straight line.
4. Nature of the graph:
- A linear equation in the form \(y = mx + b\) represents a straight line on the Cartesian plane.
- This line extends infinitely in both directions and contains an infinite number of points.
- Each point \((x, y)\) on this line is a solution to the equation \(y = -3x + 4\).
5. Matching to options:
- Option A states "a line that shows the set of all solutions to the equation". This correctly describes the graph of a linear equation.
- Option B states "a point that shows one solution to the equation", which is incorrect because the graph represents more than one solution.
- Option C states "a point that shows the \(y\)-intercept", which is misleading because while the y-intercept is a point on the graph, the entire graph is a line, not just a point.
- Option D states "a line that shows only one solution to the equation", which is incorrect because a line contains infinitely many points, each representing a solution.
Therefore, based on the analysis, the correct answer is [tex]\( \boxed{\text{A}} \)[/tex].
