Which of the points on the graph is a solution to the simultaneous inequalities y>3x+1 and y≤2x+5? Write the x-value first and the y-value second.

Answer 1

To find the solution to the simultaneous inequalities, we need to identify the region that satisfies both inequalities.

First, let's graph the line y = 3x + 1:

Starting at the y-intercept (0,1), we can use the slope by going up 3 units and to the right 1 unit to get another point. Plotting these points and drawing the line, we have:

Graph for y > 3x + 1:
```
.........
.........
.........
.........
.___
0...1
```

Now let's graph the line y = 2x + 5:

Starting at the y-intercept (0,5), we can use the slope by going up 2 units and to the right 1 unit to get another point. Plotting these points and drawing the line, we have:

Graph for y ≤ 2x + 5:
```
.........
.........
.___
0...5
```

Next, let's shade the region that satisfies both inequalities:

Since y > 3x + 1, we need to shade the region above the line y = 3x + 1.

Since y ≤ 2x + 5, we need to shade the region below or on the line y = 2x + 5.

The shaded region would be the intersection of both shaded regions, which is the region above the line y = 3x + 1 but below or on the line y = 2x + 5.

Shaded region:
```
.___
...
......
......
```

Now, let's identify the points on the graph that lie within this shaded region.

The points that satisfy both inequalities are (0, 1), (1, 2), (2, 4), and (3, 7).

Written with x-value first and y-value second, these points are:
(0, 1), (1, 2), (2, 4), (3, 7)