hello i need help with system of equations theres 3 only thank you
2x+3y-17
3x+y=31
2(x-3y)=24
y=3x-2
2(x+4)=24
x+y=17
Ah, hello there! Looks like you need some help with a few systems of equations. Let's tackle them one by one, shall we?
Equation Set 1:
2x + 3y = 17 ...(1)
3x + y = 31 ...(2)
To solve this, we can use the substitution method. In equation (2), let's solve y in terms of x:
y = 31 - 3x
Now, substitute this value of y in equation (1):
2x + 3(31 - 3x) = 17
2x + 93 - 9x = 17
-7x = -76
x = -76 / -7
x ≈ 10.857
Plug the value of x into equation (2) to find y:
3(10.857) + y = 31
32.571 + y = 31
y ≈ -1.571
Solution: x ≈ 10.857, y ≈ -1.571
Equation Set 2:
2(x - 3y) = 24 ...(3)
y = 3x - 2 ...(4)
In this case, let's solve equation (4) for x:
x = (y + 2) / 3
Substitute this value of x into equation (3):
2(((y + 2) / 3) - 3y) = 24
((y + 2) / 3) - 6y = 12
(y + 2) / 3 - 6y = 12
(y + 2) - 18y = 36
-17y = 34
y = 34 / -17
y = -2
Now, substitute the value of y into equation (4) to find x:
x = ( -2 + 2) / 3
x = 0
Solution: x = 0, y = -2
Equation Set 3:
2(x + 4) = 24 ...(5)
x + y = 17 ...(6)
Let's solve equation (5) for x:
2(x + 4) = 24
2x + 8 = 24
2x = 16
x = 16 / 2
x = 8
Now, substitute the value of x into equation (6):
8 + y = 17
y = 17 - 8
y = 9
Solution: x = 8, y = 9
I hope these responses bring a smile to your face along with the solutions! Let me know if you need any further assistance.
I can help you with solving the system of equations step-by-step. Let's take them one by one.
1. 2x + 3y = 17
3x + y = 31
To eliminate one variable, we'll multiply the second equation by 3 and the first equation by 1:
2. 2x + 3y = 17
9x + 3y = 93
Now, we'll subtract equation (1) from equation (2) to eliminate 'y':
3. (9x + 3y) - (2x + 3y) = 93 - 17
7x = 76
Dividing both sides of the equation by 7, we get:
4. x = 76 / 7
Now, substitute the value of 'x' into one of the original equations, let's use equation (1):
5. 2x + 3y = 17
2(76 / 7) + 3y = 17
152 / 7 + 3y = 17
Next, we'll simplify the equation by multiplying both sides by 7:
6. 152 + 21y = 119
Now, subtract 152 from both sides:
7. 21y = 119 - 152
21y = -33
Dividing both sides by 21, we get:
8. y = -33 / 21
So, the solution to the system of equations is:
x = 76/7
y = -33/21
Moving on to the next set of equations:
9. 2(x - 3y) = 24
y = 3x - 2
We'll start by solving equation (9) for 'x':
10. 2x - 6y = 24
2x = 24 + 6y
x = (24 + 6y) / 2
x = 12 + 3y
Now, substitute this value of 'x' into equation (10):
11. y = 3x - 2
y = 3(12 + 3y) - 2
y = 36 + 9y - 2
-8y = -34
y = -34 / -8
y = 17/4
Substitute the value of 'y' back into equation (10):
12. x = 12 + 3(17/4)
x = 12 + 51/4
x = 48/4 + 51/4
x = 99/4
Therefore, the solution to the second set of equations is:
x = 99/4
y = 17/4
I hope this helps! Let me know if you have any further questions.
Of course! I can help you solve these systems of equations. I will explain each step so that you understand the process. Let's start with the first system:
1) 2x + 3y = 17
2) 3x + y = 31
To solve this system, we can use the method of substitution.
Step 1: Solve one equation for one variable in terms of the other. Let's solve equation 2) for y:
y = 31 - 3x
Step 2: Substitute the expression of y from step 1 into the other equation (equation 1):
2x + 3(31 - 3x) = 17
Step 3: Simplify and solve for x:
2x + 93 - 9x = 17
-7x + 93 = 17
-7x = 17 - 93
-7x = -76
x = -76 / -7
x ≈ 10.857
Step 4: Substitute the value of x into one of the original equations (let's use equation 2) to solve for y:
3(10.857) + y = 31
32.571 + y = 31
y = 31 - 32.571
y ≈ -1.571
So, the solution to the first system of equations is approximately x = 10.857 and y = -1.571.
Now, let's move on to the second system:
1) 2(x - 3y) = 24
2) y = 3x - 2
Let's solve this system using the method of substitution as well.
Step 1: Substitute the expression for y from equation 2) into equation 1):
2(x - 3(3x - 2)) = 24
Step 2: Simplify and solve for x:
2(x - 9x + 6) = 24
2(-8x + 6) = 24
-16x + 12 = 24
-16x = 24 - 12
-16x = 12
x = 12 / -16
x = -0.75
Step 3: Substitute the value of x back into one of the original equations (let's use 2) to solve for y:
y = 3(-0.75) - 2
y = -2.25 - 2
y = -4.25
Therefore, the solution to the second system of equations is x = -0.75 and y = -4.25.
Finally, let's solve the third system:
1) 2(x + 4) = 24
2) x + y = 17
We'll solve this system using the method of elimination.
Step 1: Simplify equation 1:
2x + 8 = 24
2x = 24 - 8
2x = 16
x = 16 / 2
x = 8
Step 2: Substitute the value of x into equation 2:
8 + y = 17
y = 17 - 8
y = 9
Hence, the solution to the third system of equations is x = 8 and y = 9.
I hope this helps! Let me know if you have any further questions.