How do I solve the following problems?
My homework was to correct a puiz that I failed, but I do not know how to solve these problems..please help!
Solving systems of equations using elimination:
7.) 10x - 10y = 0
5x - 4y = 8
Answer (8,8)
I got the right answer because this one was multiple choice, but it was because I guessed...I need to know how to work it out.
8.) -8x - 8y = -8
-5x + 16y = 16
9.)I need this one worked out...I guessed the right answer (7, -7)
2x + 3yy = -7
-3x - 2y = -7
10.) -8x + 8y = -16
-3x + 7y = 10
11. 8x + 14y = 128
6x + 2y = 62
#7:
if 10x-10y=0, then 10x=10y, or x=y
plug that into the other equation to get
5x-4x = 8
x=8, so y=8
the others are basically the same. you can substitute, or eliminate.
#8:
-8x - 8y = -8
-5x + 16y = 16
x+y=1, so y = 1-x
-5x + 16(1-x) = 16
-21x = 0
x=0, so y=1
elimination:
#10
-8x + 8y = -16
-3x + 7y = 10
multiply top by 3 and bottom by -8 to get
-24x + 24y = -48
24x - 56y = -80
add them up to get
-32y = -128
y = 4, so x = 6
Try the others and come back if you get stuck. Show us whatcha got, and we can straighten it out if necessary.
If you get an answer, be sure to plug the values back into both original equations to make sure it is right.
To solve systems of equations using elimination, you can follow these steps:
Step 1: Choose one of the equations and multiply both sides by a number that will result in the same coefficient for one of the variables when added or subtracted to the corresponding term in the other equation. This allows for elimination of one of the variables when the two equations are added or subtracted.
Step 2: Add or subtract the two equations to eliminate one of the variables. This will result in a new equation with only one variable.
Step 3: Solve the new equation to find the value of the remaining variable.
Step 4: Substitute the value of the variable found in Step 3 back into one of the original equations to solve for the other variable.
Now, let's apply these steps to the problems you provided:
7)
Equation 1: 10x - 10y = 0
Equation 2: 5x - 4y = 8
Let's eliminate the variable "x" by multiplying Equation 2 by 2, which will result in the same coefficient as in Equation 1:
Equation 1: 10x - 10y = 0
Equation 2 (multiplied by 2): 10x - 8y = 16
Now, subtract Equation 2 from Equation 1:
(10x - 10y) - (10x - 8y) = 0 - 16
-2y = -16
Divide both sides by -2:
y = 8
Substitute the value of y = 8 into Equation 1:
10x - 10(8) = 0
10x - 80 = 0
10x = 80
Divide both sides by 10:
x = 8
So, the solution to the system of equations is (8, 8).
8)
Equation 1: -8x - 8y = -8
Equation 2: -5x + 16y = 16
To eliminate the variable "y", let's multiply Equation 1 by -2 and Equation 2 by 1:
Equation 1 (multiplied by -2): 16x + 16y = 16
Equation 2 (multiplied by 1): -5x + 16y = 16
Now, add Equation 1 and Equation 2:
(16x + 16y) + (-5x + 16y) = 16 + 16
11x + 32y = 32
This new equation still has two variables, so we can't solve it. It means the system of equations has no unique solution. The lines represented by the equations are parallel, and they never intersect.
9)
Equation 1: 2x + 3yy = -7
Equation 2: -3x - 2y = -7
To eliminate the variable "y", let's multiply Equation 1 by 2 and Equation 2 by 3:
Equation 1 (multiplied by 2): 4x + 6yy = -14
Equation 2 (multiplied by 3): -9x - 6y = -21
Now, add Equation 1 and Equation 2:
(4x + 6yy) + (-9x - 6y) = -14 - 21
-5x = -35
Divide both sides by -5:
x = 7
Substitute the value of x = 7 into Equation 1:
2(7) + 3yy = -7
14 + 3yy = -7
3yy = -21
Divide both sides by 3:
yy = -7
y = -7
So, the solution to the system of equations is (7, -7).
10)
Equation 1: -8x + 8y = -16
Equation 2: -3x + 7y = 10
To eliminate the variable "x", let's multiply Equation 1 by 3 and Equation 2 by 8:
Equation 1 (multiplied by 3): -24x + 24y = -48
Equation 2 (multiplied by 8): -24x + 56y = 80
Now, subtract Equation 1 from Equation 2:
(-24x + 56y) - (-24x + 24y) = 80 - (-48)
80y = 128
Divide both sides by 80:
y = 128/80
y = 1.6
Substitute the value of y = 1.6 into Equation 1:
-8x + 8(1.6) = -16
-8x + 12.8 = -16
-8x = -28.8
Divide both sides by -8:
x = 3.6
So, the solution to the system of equations is (3.6, 1.6).
11)
Equation 1: 8x + 14y = 128
Equation 2: 6x + 2y = 62
To eliminate the variable "y", let's multiply Equation 1 by -2 and Equation 2 by 7:
Equation 1 (multiplied by -2): -16x - 28y = -256
Equation 2 (multiplied by 7): 42x + 14y = 434
Now, add Equation 1 and Equation 2:
(-16x - 28y) + (42x + 14y) = -256 + 434
26x = 178
Divide both sides by 26:
x = 178/26
x = 6.846
Substitute the value of x = 6.846 into Equation 1:
8(6.846) + 14y = 128
54.768 + 14y = 128
14y = 128 - 54.768
14y = 73.232
Divide both sides by 14:
y = 73.232/14
y = 5.23
So, the solution to the system of equations is (6.846, 5.23).
Remember, these steps can be used to solve any system of equations using elimination. Make sure to carefully follow the steps and perform the calculations accurately to obtain the correct solution.