how i can soliving
1)x>=0
y>=0
y<=3x+3
y<=-x+7
2)x>=0
y>=-1
y<=x+1
y<=-1/4x+6
To solve these systems of inequalities, we need to find the region on a graph where all the conditions are simultaneously satisfied. Here's how you can solve each of these systems step-by-step:
1) System of inequalities:
x >= 0
y >= 0
y <= 3x + 3
y <= -x + 7
Step 1: Graph the boundary lines:
Start by graphing the boundary lines for each inequality. Replace the inequality signs with an equal sign, and draw the corresponding line on the graph.
For the first inequality, x >= 0, draw a vertical line passing through x = 0.
For the second inequality, y >= 0, draw a horizontal line passing through y = 0.
For the third inequality, y <= 3x + 3, rearrange the equation to y = 3x + 3. This equation represents a line with a slope of 3 and a y-intercept of 3. Plot this line.
For the fourth inequality, y <= -x + 7, rearrange the equation to y = -x + 7. This equation represents a line with a slope of -1 and a y-intercept of 7. Plot this line.
Step 2: Determine the shaded region:
To find the region that satisfies all inequalities, determine the shaded area that overlaps. Shade the area common to all regions.
In this case, the shaded region will be the area below both lines (3x + 3 and -x + 7), above the x-axis, and to the right of the y-axis.
Step 3: Identify the solution set:
Look at the shaded region and identify the points that are within this region. These points satisfy all the given inequalities.
In this case, the solution set will include all points in the shaded region.
2) System of inequalities:
x >= 0
y >= -1
y <= x + 1
y <= -1/4x + 6
Repeat the same steps as in the previous example:
Step 1: Graph the boundary lines:
Graph the boundary lines for each inequality by replacing the inequality signs with an equal sign and drawing the corresponding line.
For the first inequality, x >= 0, draw a vertical line passing through x = 0.
For the second inequality, y >= -1, draw a horizontal line passing through y = -1.
For the third inequality, y <= x + 1, rearrange the equation to y = x + 1. This represents a line with a slope of 1 and a y-intercept of 1. Plot this line.
For the fourth inequality, y <= -1/4x + 6, rearrange the equation to y = -1/4x + 6. This represents a line with a slope of -1/4 and a y-intercept of 6. Plot this line.
Step 2: Determine the shaded region:
Determine the shaded region that satisfies all the inequalities. Shade the common area.
In this case, the shaded region will be the area below both lines (x + 1 and -1/4x + 6), above the line y = -1, and to the right of the y-axis.
Step 3: Identify the solution set:
Look at the shaded region and identify the points within this region. These points satisfy all the given inequalities.
In this case, the solution set will include all points in the shaded region.
Remember, graphing the inequalities is an effective way to visualize the solution set. However, if you need to find the exact coordinates of the solution set, you might need to use additional techniques like substitution or elimination.