Answer:
a. Let's solve this system of equations:
1. b = 13 - 2a
2. -1 = 3b - 4a
First, substitute the expression for b from equation 1 into equation 2:
-1 = 3(13 - 2a) - 4a
Now, simplify and solve for a:
-1 = 39 - 6a - 4a
-1 = 39 - 10a
-1 - 39 = -10a
-40 = -10a
Divide both sides by -10 to solve for a:
a = 4
Now that we have the value of a, we can find b using equation 1:
b = 13 - 2a
b = 13 - 2(4)
b = 13 - 8
b = 5
So, the solution to the system of equations is a = 4 and b = 5.
b. Let's solve this system of equations:
1. 3c + 2d = 4
2. 4d = 90 + 13
First, solve equation 2 for d:
4d = 103
d = 103 / 4
d = 25.75
Now, substitute the value of d into equation 1:
3c + 2(25.75) = 4
3c + 51.5 = 4
3c = 4 - 51.5
3c = -47.5
c = -47.5 / 3
c = -15.83 (rounded to two decimal places)
So, the solution to the system of equations is c ≈ -15.83 and d ≈ 25.75.
c. Let's solve this system of equations:
1. 1 - 3n = 2m
2. 12m - 6 = -18n
First, solve equation 1 for m:
1 - 3n = 2m
2m = 1 - 3n
m = (1 - 3n) / 2
Now, substitute the expression for m into equation 2:
12[(1 - 3n) / 2] - 6 = -18n
6(1 - 3n) - 6 = -18n
6 - 18n - 6 = -18n
0 = 0
The system of equations is dependent, meaning there are infinitely many solutions, and they can be expressed as:
m = (1 - 3n) / 2
where n is any real number.
d. Let's solve this system of equations:
1. x + y - 2z = 1
2. 2x - 3y + 5z = 7
3. -x + 2y - 3z = -4
We can use the method of substitution to solve this system. First, solve equation 1 for x:
x = 1 - y + 2z
Now, substitute this expression for x into equations 2 and 3:
2(1 - y + 2z) - 3y + 5z = 7
-(1 - y + 2z) + 2y - 3z = -4
Now, simplify each equation:
1. 2 - 2y + 4z - 3y + 5z = 7
2. -1 + y - 2z + 2y - 3z = -4
Simplify further:
1. -5y + 9z = 5
2. 3y - 5z = -3
Now, let's solve this system of two equations:
3. -5y + 9z = 5
4. 3y - 5z = -3
First, multiply equation 4 by 3 to make the coefficients of y in both equations equal:
9y - 15z = -9
Now, add equation 3 and the modified equation 4:
(-5y + 9z) + (9y - 15z) = 5 - 9
4y - 6z = -4
Now, solve for y:
4y = -4 + 6z
y = (-4 + 6z) / 4
y = (-2 + 3z) / 2
Now that we have expressions for y and x, we can express the solution in terms of z:
x = 1 - y + 2z
x = 1 - (-2 + 3z) / 2 + 2z
x = (5z + 3) / 2
So, the solution to the system of equations is:
x = (5z + 3) / 2
y = (-2 + 3z) / 2
z is any real number.