a + b = 6, b + c = 20, and c + a = 7.5, solve for a , b and c.
3+3=6
10+10=20
4.3+3.3
equation #2 - equation #3:
b - a = 18.5
add this to equation #1
2b = 18.5
b = 9.25
put this into #1
a + 9.25 = 6
a = -3.25
put this into #3
c - 3.25 = 7.5
c = 10.75
One method:
Use substitution to express an equation in one unknown.
Choose a.
From the first equation...
a + b = 6
So, b = 6 - a
From the third equation...
c + a = 7.5
So, c = 7.5 - a
Now write the second equation substituting the values for b and c in terms of a:
b + c = 20
From the above values for b and c in terms of a:
(6 - a) + (7.5 - a) = 20
6 - a + 7.5 - a = 20
13.5 - 2a = 20
-2a = 6.5
Solve for a.
Substitute that to find b and c.
To solve for a, b, and c in the given equations, we can use a method called substitution.
Given equations:
1) a + b = 6
2) b + c = 20
3) c + a = 7.5
Let's solve for a first:
From equation 3), we have c + a = 7.5.
Solving for a, we get: a = 7.5 - c.
Now, substitute the value of a in equations 1) and 2) to solve for b and c:
Substituting a = 7.5 - c in equation 1), we have:
(7.5 - c) + b = 6.
Rearranging the equation, we get: b = 6 - 7.5 + c.
Simplifying it further, we have: b = -1.5 + c.
Now, substitute a = 7.5 - c in equation 2):
b + c = 20.
Substituting the value of b, we have:
(-1.5 + c) + c = 20.
Simplifying it further, we get: 2c - 1.5 = 20.
Adding 1.5 to both sides of the equation, we have: 2c = 21.5.
Dividing both sides by 2, we get: c = 10.75.
Now, substitute the value of c in equation b = -1.5 + c:
b = -1.5 + 10.75.
Simplifying it, we have: b = 9.25.
Finally, substitute the values of b and c in equation a = 7.5 - c:
a = 7.5 - 10.75.
Simplifying it, we have: a = -3.25.
So, the solutions are:
a = -3.25,
b = 9.25,
c = 10.75.