Solve the simultaneous equation
n/4-m/6=-1
m/5+n/2=10
Using elimination method
Is this the elimination method?
Multiply by 24.
6n - 4m = -24
Multiply by 10.
2m + 5n = 100
Multiply second equation by 2.
10n + 4m = 200
Add the two equations.
16n = 76
To solve the simultaneous equations using the elimination method, we will eliminate one variable by manipulating the equations.
Let's start by multiplying both sides of the first equation by 6 to get rid of the fractions:
6 * (n/4) - 6 * (m/6) = 6 * (-1)
3n - 2m = -6 (equation 1)
Now, multiply both sides of the second equation by 4 to eliminate the fractions:
4 * (m/5) + 4 * (n/2) = 4 * 10
(4/5)m + 2n = 40 (equation 2)
Now, we have two equations with no fractions:
3n - 2m = -6 (equation 1)
(4/5)m + 2n = 40 (equation 2)
To eliminate the n variable, we will multiply equation 1 by 2 and equation 2 by 3:
2(3n - 2m) = 2(-6)
3(4/5)m + 3(2n) = 3(40)
Simplifying these equations, we get:
6n - 4m = -12 (equation 3)
(12/5)m + 6n = 120 (equation 4)
Now, we can subtract equation 3 from equation 4 to eliminate the n variable:
((12/5)m + 6n) - (6n - 4m) = 120 - (-12)
(12/5)m + 6n - 6n + 4m = 120 + 12
(12/5)m + 4m = 132
(12m + 20m) / 5 = 132
32m / 5 = 132
To solve for m, let's multiply both sides by 5/32:
(5/32) * (32m / 5) = (5/32) * 132
m = 660/32
m = 20.625
Now, substitute the value of m back into one of the original equations (let's use equation 1) to solve for n:
3n - 2(20.625) = -6
3n - 41.25 = -6
3n = -6 + 41.25
3n = 35.25
n = 35.25/3
n ≈ 11.75
Therefore, the solution to the simultaneous equations is:
n ≈ 11.75
m ≈ 20.625
To solve the simultaneous equation using the elimination method, we need to eliminate one of the variables by either multiplying the two equations or modifying them in a way that will result in eliminating one variable when the two equations are added or subtracted.
Let's begin by eliminating the m variable.
First equation: n/4 - m/6 = -1 (equation 1)
Second equation: m/5 + n/2 = 10 (equation 2)
To eliminate the m variable, we can multiply equation 1 by 5 and equation 2 by 6 to create common denominators for the m term:
5 * (n/4 - m/6) = 5 * (-1)
6 * (m/5 + n/2) = 6 * 10
This simplifies to:
5n/4 - 5m/6 = -5
6m/5 + 6n/2 = 60
Now, we have the following two equations:
5n/4 - 5m/6 = -5 (equation 3)
6m/5 + 3n = 60 (equation 4)
Notice that equation 4 already has the m variable isolated, so we can multiply equation 4 by 5 to create a common denominator of 30:
5 * (6m/5 + 3n) = 5 * 60
This simplifies to:
6m + 15n = 300 (equation 5)
Now we have the following two equations:
5n/4 - 5m/6 = -5 (equation 3)
6m + 15n = 300 (equation 5)
We can further simplify equation 3 by multiplying through by 12 to get rid of the fractions:
12 * (5n/4 - 5m/6) = 12 * (-5)
This simplifies to:
15n - 10m = -60 (equation 6)
Now we have the following two equations:
15n - 10m = -60 (equation 6)
6m + 15n = 300 (equation 5)
To eliminate the n variable, let's multiply equation 6 by 3 and equation 5 by 2:
3 * (15n - 10m) = 3 * (-60)
2 * (6m + 15n) = 2 * 300
This simplifies to:
45n - 30m = -180 (equation 7)
12m + 30n = 600 (equation 8)
Now we have the following two equations:
45n - 30m = -180 (equation 7)
12m + 30n = 600 (equation 8)
We can now eliminate the n variable by multiplying equation 7 by 2:
2 * (45n - 30m) = 2 * (-180)
This simplifies to:
90n - 60m = -360 (equation 9)
Now we have the following two equations:
90n - 60m = -360 (equation 9)
12m + 30n = 600 (equation 8)
We can eliminate the m variable now by multiplying equation 9 by 6:
6 * (90n - 60m) = 6 * (-360)
This simplifies to:
540n - 360m = -2160 (equation 10)
Now we have the following two equations:
540n - 360m = -2160 (equation 10)
12m + 30n = 600 (equation 8)
We can add these two equations to eliminate the m variable:
(540n - 360m) + (12m + 30n) = -2160 + 600
This simplifies to:
570n = -1560
Now, we can solve for n by dividing both sides of the equation by 570:
n = -1560 / 570
Simplifying further gives:
n = -52/19
Finally, to find the value of m, we can substitute the value of n back into equation 8:
12m + 30 * (-52/19) = 600
Simplifying this equation will give the value of m.
Note: Working with fractions can be challenging, so make sure to do the calculations carefully to avoid errors.