Can I get help in solving these problems using the linear system of elimination.

4x - 5y =22
x + 2y = -1

2x - 3y = 16
3x + 4y = 7

Word Problem:
During the summer, you want to earn at least $150 per week. You earn $10 per hour working for a farmer and you earn $5 per hour babysitting for your neighbor. You can work at most 25 hours per week.

I need to write and solve a system of linear inequalities that models the situation. Let x be the number of hours per week working on the farm and let y be the number of hours per week babysitting...

I am not sure but I came up with x+y<=(less than equal) to 25. or is there another solution....Thanks

For your first question, substitute x = -1 -2y into the first equation, leaving you with

-4 -8y -5y = 22, which cqn be rearranged to
-13y = 26
y = -2
2x +6 = 16
2x = 10
x=5

Do your second question the same way.

For your third question, the other inequality you have is
10 x + 5 y >= 150
Solve the "equals" case first.
x + y = 25
10x + 5y = 150
5x + 5y = 125
5x = 25
x = 5
y = 20
If you work more than 5 hours per week at the higher wage, and 20 or less at the lower paying job (but totaling 25 hours) you will earn more than $150 per week. Plot a graph to show the allowed regions of x-y space.

To solve the given system of equations using the elimination method:

1) 4x - 5y = 22
x + 2y = -1

Step 1: Multiply the second equation by 4 to make the coefficients of x in both equations equal.
4(x + 2y) = 4(-1)
4x + 8y = -4

Step 2: Now, we can eliminate the x term by subtracting the second equation from the first equation.
(4x - 5y) - (4x + 8y) = 22 - (-4)
4x - 5y - 4x - 8y = 22 + 4
-13y = 26

Step 3: Divide both sides of the equation by -13 to solve for y.
y = 26 / -13
y = -2

Step 4: Substitute the value of y back into the second equation to solve for x.
x + 2(-2) = -1
x - 4 = -1
x = -1 + 4
x = 3

Therefore, the solution to the system of equations is x = 3 and y = -2.

2) 2x - 3y = 16
3x + 4y = 7

Step 1: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x in both equations equal.
3(2x - 3y) = 3(16)
6x - 9y = 48
2(3x + 4y) = 2(7)
6x + 8y = 14

Step 2: Now, we can eliminate the x term by subtracting the first equation from the second equation.
(6x + 8y) - (6x - 9y) = 14 - 48
6x + 8y - 6x + 9y = 14 - 48
17y = -34

Step 3: Divide both sides of the equation by 17 to solve for y.
y = -34 / 17
y = -2

Step 4: Substitute the value of y back into the first equation to solve for x.
2x - 3(-2) = 16
2x + 6 = 16
2x = 16 - 6
2x = 10
x = 10 / 2
x = 5

Therefore, the solution to the system of equations is x = 5 and y = -2.

For the word problem:

Let x be the number of hours per week working on the farm,
and let y be the number of hours per week babysitting.

To write the system of linear inequalities that models the situation:

1) You want to earn at least $150 per week, so the total earnings from working on the farm and babysitting combined should be greater than or equal to $150.
10x + 5y ≥ 150

2) You earn $10 per hour working for the farmer, so the total number of hours for farming should be greater than or equal to zero and less than or equal to 25.
0 ≤ x ≤ 25

3) You earn $5 per hour babysitting, so the total number of hours for babysitting should be greater than or equal to zero and less than or equal to 25.
0 ≤ y ≤ 25

Combining all three inequalities, the system can be represented as:
10x + 5y ≥ 150
0 ≤ x ≤ 25
0 ≤ y ≤ 25

To solve the system of linear equations using the method of elimination, follow these steps:

1. Rearrange the equations if necessary to eliminate one variable. In this case, we don't need to rearrange the equations.

2. Choose one equation and multiply it by a constant so that the coefficients of one of the variables will cancel out when added or subtracted. In this case, let's choose to eliminate the variable x.

Multiply the second equation by 4:
4(x + 2y) = 4(-1)
4x + 8y = -4

3. Add or subtract the modified equations to eliminate one of the variables:

(4x - 5y) + (4x + 8y) = 22 + (-4)
8x + 3y = 18

4. Solve the resulting equation for one variable. Here, let's solve for x:
8x = 18 - 3y
x = (18 - 3y) / 8

5. Substitute the value of x back into one of the original equations to solve for the other variable. Let's substitute it into the first equation:
4((18 - 3y) / 8) - 5y = 22
(9 - (3/8)y) - 5y = 22
9 - (3/8)y - 5y = 22
- (43/8)y = 13

6. Solve for y:
(43/8)y = -13
y = (-13 * 8) / 43
y = -104 / 43

7. Finally, substitute the value of y back into the equation for x to find its value:
x = (18 - 3(-104/43)) / 8
x = (18 + 312/43) / 8
x = (774/43) / 8
x = 774 / (43 * 8)

Therefore, the solution to the system of equations is:
x = 774 / (43 * 8) and y = -104 / 43

For the word problem, let's break it down:

1. We want to earn at least $150 per week.
The amount earned working for the farmer is $10 per hour, which we will represent as 10x.
The amount earned babysitting is $5 per hour, represented as 5y.
Thus, the total amount earned per week is 10x + 5y.

2. We can work at most 25 hours per week.
This can be represented as x + y ≤ 25.

3. Putting it all together, we can write the system of linear inequalities as:
10x + 5y ≥ 150 (earning at least $150 per week)
x + y ≤ 25 (working at most 25 hours per week)

These two inequalities model the situation described in the word problem.