3p-5q=5p+8=2p+q+7
Solve this with elimination method
To solve this system of equations using the elimination method, we will try to eliminate one variable by combining two of the equations.
First, let's eliminate the q variable. We can do this by multiplying the third equation by -5 and adding it to the first equation, which gives:
3p - 5q + (-5)(2p + q + 7) = 5p + 8
Simplifying this equation, we get:
3p - 5q - 10p - 5q - 35 = 5p + 8
Combining like terms, we have:
-7p - 10q - 35 = 5p + 8
Next, we can eliminate the p variable by multiplying the second equation by -3 and adding it to the third equation:
(-3)(3p - 5q) = -3(5p + 8)
Simplifying this equation, we get:
-9p + 15q = -15p - 24
To eliminate the p variable, we can multiply the above equation by -7 and add it to the equation -7p - 10q - 35 = 5p + 8:
-7(-9p + 15q) + (-7p - 10q - 35) = 0
63p - 105q + 7p + 10q + 35 = 0
Combining like terms, we have:
70p - 95q + 35 = 0
Now, we have two equations: -9p + 15q = -15p - 24 and 70p - 95q + 35 = 0.
We can solve this system of equations using the substitution method or the elimination method again.
Let's use the elimination method again to eliminate the p variable by multiplying the first equation by 7 and the second equation by 9:
7(-9p + 15q) = 7(-15p - 24)
63p - 105q = -105p - 168
9(70p - 95q + 35) = 9(0)
630p - 855q + 315 = 0
Now, we have two equations: 63p - 105q = -105p - 168 and 630p - 855q + 315 = 0.
We can also use the elimination method to eliminate the p variable by multiplying the first equation by 6 and the second equation by 1:
6(63p - 105q) = 6(-105p - 168)
378p - 630q = -630p - 1008
1(630p - 855q + 315) = 1(0)
630p - 855q + 315 = 0
Now, we have two equations: 378p - 630q = -630p - 1008 and 630p - 855q + 315 = 0.
We can solve this system of equations using the substitution method or the elimination method again.
Using the elimination method one more time, we can subtract the equation 630p - 855q + 315 = 0 from the equation 378p - 630q = -630p - 1008:
(378p - 630q) - (630p - 855q + 315) = (-630p - 1008) - 0
378p - 630q - 630p + 855q - 315 = -630p - 1008
-252p + 225q - 315 = -630p - 1008
Combining like terms, we have:
378q = -378p - 693
q = (-378p - 693) / 378
Now, we can substitute this value of q into one of the original equations to solve for p. Let's use the first equation:
3p - 5q = 5p + 8
Substituting q = (-378p - 693) / 378 in the equation, we have:
3p - 5((-378p - 693) / 378) = 5p + 8
Simplifying this equation, we get:
3p - (-1890p - 3465) / 378 = 5p + 8
Multiplying through by 378 to get rid of the fraction:
1134p - (-1890p - 3465) = 1890p + 3024
Distributing the negative sign:
1134p + 1890p + 3465 = 1890p + 3024
Combining like terms:
3024p + 3465 = 1890p + 3024
Subtracting 1890p from both sides:
3024p + 3465 - 1890p = 1890p + 3024 - 1890p
1134p + 3465 = 3024
Subtracting 3465 from both sides:
1134p + 3465 - 3465 = 3024 - 3465
1134p = -441
Dividing both sides by 1134:
p = -441 / 1134
Simplifying this fraction, we have:
p ≈ -0.389
Substituting this value of p back into the equation q = (-378p - 693) / 378, we can solve for q:
q = (-378(-0.389) - 693) / 378
Simplifying this equation, we get:
q ≈ -0.059
Therefore, the solution to the system of equations is p ≈ -0.389 and q ≈ -0.059.