In Exercises 3-6: Only the ordered pair (1, 1) is a solution to the system of inequalities.
In Exercises 7-10: The solutions to the system of inequalities are (-5, 2) and (1, -1).
In Exercises 21-26: 21.
21. The system of linear equations is: y = -2x + 5, y = x - 3
22. The system of linear equations is: y = -2x + 4, y = x - 3
23. The system of linear equations is: y = -x + b₁, y = 2x + b₂
24. The system of linear equations is: y = -x + 7, y = 2x - 3
25. The system of linear equations is: y = -x - 3, y = (2/3)x - 5
26. The system of linear equations is: y = x + 3, y = (-3/2)x + 5
In Exercises 3-6:
1. Graph each inequality:
Inequality 1: 4x + 3y ≤ 12
Convert the inequality to slope-intercept form (y = mx + b) to find the y-intercept: y ≤ (-4/3)x + 4
Plot the y-intercept at (0, 4).
Since the inequality is ≤, shade the region below the line.
Inequality 2: 2x - y ≥ -5
Convert the inequality to slope-intercept form: y ≤ 2x + 5
Plot the y-intercept at (0, 5).
Since the inequality is ≤, shade the region below the line.
2. Find the intersection of the shaded regions:
The solution to the system of inequalities is the area where both shaded regions overlap.
3. Check the ordered pairs: (-4, 3):
Substitute x = -4 and y = 3 into both inequalities:
4(-4) + 3(3) ≤ 12 --> -9 ≤ 12 (true)
2(-4) - 3 ≤ -5 --> -1 ≤ -5 (false)
Since one inequality is not true, (-4, 3) is not a solution.
(-3, -1):
Substitute x = -3 and y = -1 into both inequalities:
4(-3) + 3(-1) ≤ 12 --> -15 ≤ 12 (false)
2(-3) - (-1) ≤ -5 --> 5 ≤ -5 (false)
Since both inequalities are not true, (-3, -1) is not a solution.
(-2, 5):
Substitute x = -2 and y = 5 into both inequalities:
4(-2) + 3(5) ≤ 12 --> 7 ≤ 12 (true)
2(-2) - 5 ≤ -5 --> 1 ≤ -5 (false)
Since one inequality is not true, (-2, 5) is not a solution (1, 1):
Substitute x = 1 and y = 1 into both inequalities:
4(1) + 3(1) ≤ 12 --> 7 ≤ 12
2(1) - 1 ≤ -5 --> -3 ≤ -5
Since both inequalities are true, (1, 1) is a solution.
7. (-5, 2): Check if (-5, 2) satisfies the first inequality: y < 4. In this case, 2 < 4, so yes.
Check if (-5, 2) satisfies the second inequality: x + 3 > y. In this case, -5 + 3 > 2, so yes.
Since both inequalities are satisfied, (-5, 2) is a solution.
8. (1, -1): Check if (1, -1) satisfies the first inequality: y > -2. In this case, -1 > -2, so yes.
Check if (1, -1) satisfies the second inequality: y > x - 5. In this case, -1 > 1 - 5, so yes.
Since both inequalities are satisfied, (1, -1) is a solution.
9. (0, 0): Check if (0, 0) satisfies the first inequality: y ≤ x + 7. In this case, 0 ≤ 0 + 7, so yes.
Check if (0, 0) satisfies the second inequality: y ≥ 2x + 3. In this case, 0 ≥ 2(0) + 3, so no.
Since only one inequality is satisfied, (0, 0) is not a solution.
10. (4, -3): Check if (4, -3) satisfies the first inequality: y > 5 - x + 1. In this case, -3 > 5 - 4 + 1, so no. Since only one inequality needs to be checked and it's not satisfied, (4, -3) is not a solution.
21. Slope: -2
Y-intercept: 5
Therefore, the equation for the is: y = -2x + 5
Slope: 1
Y-intercept: -3
Therefore, the equation for the is: y = x - 3
22. This line has a slope of -2 and a y-intercept of 4. Therefore, its equation is: y = -2x + 4.
This line has a slope of 1 and a y-intercept of -3. Therefore, its equation is: y = x - 3.
23. This line has a slope of -1.
This line has a slope of 2.
Therefore, the system of linear equations is approximately:
y = -x + b₁
y = 2x + b₂
24. Slope: -1
Y-intercept: 7
Therefore, the equation for the blue line is: y = -x + 7.
Slope: 2
Y-intercept: -3
Therefore, the equation for the green line is: y = 2x - 3.
25. The system of linear equations is: y = -x - 3, y = (2/3)x - 5
26. Slope: 1
Y-intercept: 3
Therefore, the equation is:y = x + 3
Slope: -3/2
Y-intercept: 5
Therefore, the equation is: y = (-3/2)x + 5.