Graph b) best depicts the solution as it shades the region satisfying both x≤ 3 and y> -3x+1.
To solve this system of inequalities, we can first graph each inequality individually.
Graphing x ≤ 3:
This inequality represents all points that lie to the left of or on the line x = 3. To graph this, we can draw a vertical line at x = 3. The line itself represents all points where x = 3, and the region to the left of the line represents all points where x < 3. Since the inequality includes the equal sign (≤), we must also shade in the line itself.
Graphing y > -3x + 1:
This inequality represents all points that lie above the line y = -3x + 1. To graph this, we can first find the y-intercept. The y-intercept is the point where the line crosses the y-axis, which happens when x = 0. Substituting x = 0 into the equation, we get y = 1. Therefore, the line must pass through the point (0, 1).
Next, we can find the slope of the line. The slope is equal to -3, which means that for every 3 units we move down (because it's negative) on the y-axis, we must also move 1 unit to the right on the x-axis. To graph this, we can start at the y-intercept and move 3 units down and 1 unit to the right. This brings us to the point (-1, -2). We can then draw a line through the points (0, 1) and (-1, -2).
Shading the solution region:
Now that we have graphed both inequalities, we can shade the region that satisfies both inequalities. The solution region is the region that lies to the left of the line x = 3 and above the line y = -3x + 1. This region is shaded in blue in graph b).
Therefore, the graph that best shows the solution to the system of inequalities is b).
Here is a justification for my answer:
Graph a) does not satisfy the inequality y > -3x + 1. The shaded region in graph a) lies below the line y = -3x + 1, which means that the points in this region do not satisfy the inequality.
Graph c) does not satisfy the inequality x ≤ 3. The shaded region in graph c) includes points where x > 3, which means that these points do not satisfy the inequality.
Graph d) does not satisfy either inequality. The shaded region in graph d lies below the line y = -3x + 1 and includes points where x > 3.
Therefore, the only graph that satisfies both inequalities is b).
The correct graph is b).