If a+b=6, b+c=−3, and a+c=5, what is the value of a+b+c?
i am not sure how to solve this but would you do this to solve it:
A+B+B+C+A+A+C=6+-3+5
well you could solve the equations for a, b and c but your way is more fun
You have too many a s though. There should be 2 of each letter
a+b + b+c + a+c = 8
2a + 2b + 2c = 8
so
a+b+c = 4
yay i take russian math and i need answers thanks!!
Well, you certainly have the right idea with your equation, but your summation seems to be a bit off. Let's give it another shot, shall we?
Since a + b = 6, we can rewrite it as a = 6 - b. Similarly, b + c = -3 can be rewritten as c = -3 - b. Finally, a + c = 5 can be rewritten as c = 5 - a.
Now we can substitute these expressions into the equation a + b + c:
(6 - b) + b + (-3 - b) = 6 - b + b - 3 - b
Simplifying this equation, we get:
6 - b + b - 3 - b = 6 - 3 - b - b
Which further simplifies to:
6 - 3 - b - b = 3 - 2b
So, the value of a + b + c is 3 - 2b.
To solve this system of equations, we can use a method called substitution. Let's go step-by-step to find the values of a, b, and c:
1. Start with the given equations:
a + b = 6 -- Equation (1)
b + c = -3 -- Equation (2)
a + c = 5 -- Equation (3)
2. We will solve Equation (1) for a:
a = 6 - b
3. Substitute the value of a in Equation (2):
(6 - b) + c = -3
4. Simplify Equation (2):
6 - b + c = -3
5. Now, we will solve Equation (3) for a:
a = 5 - c
6. Substitute the value of a in Equation (2):
(5 - c) + c = -3
7. Simplify Equation (3):
5 - c + c = -3
8. Combine like terms in Equation (3):
5 = -3
9. Since 5 ≠ -3, this implies that the system of equations has no solution.
Therefore, there is no unique solution for the values of a, b, and c, and we cannot determine the value of a + b + c.
To solve this system of equations, you want to find the values of a, b, and c that simultaneously satisfy all three equations. You are correct about adding the three equations together to simplify the problem.
Start by adding the three equations together:
a + b + b + c + a + a + c = 6 + (-3) + 5
This simplifies to:
3a + 2b + 2c = 8
Next, you need to re-arrange the equation to solve for one variable in terms of the others. In this case, let's solve for a in terms of b and c:
Subtract 2b + 2c from both sides:
3a = 8 - 2b - 2c
Divide both sides by 3:
a = (8 - 2b - 2c) / 3
Now that you have the value of a in terms of b and c, substitute this expression into any one of the original equations. Let's substitute it into the second equation b+c = -3:
(8 - 2b - 2c) / 3 + b + c = -3
Multiply both sides by 3 to eliminate the fraction:
8 - 2b - 2c + 3b + 3c = -9
Combine like terms:
b + c - 2b - 2c = -9 - 8 + 2
Simplify:
-b - c = -15
Now you have one equation with only two variables, b and c. Let's solve for b in terms of c:
Multiply both sides by -1:
b + c = 15
Subtract c from both sides:
b = 15 - c
Now that you have the value of b in terms of c, substitute this expression into any one of the original equations. Let's substitute it into the first equation a + b = 6:
a + (15 - c) = 6
Subtract 15 - c from both sides:
a = 6 - 15 + c
Combine like terms:
a = -9 + c
Now you have the value of a in terms of c. Substitute this expression into any one of the original equations. Let's substitute it into the third equation a + c = 5:
(-9 + c) + c = 5
Combine like terms:
-9 + 2c = 5
Add 9 to both sides:
2c = 14
Divide both sides by 2:
c = 7
Now that you have the value of c, substitute it back into the expression for b:
b = 15 - c
b = 15 - 7
b = 8
Finally, substitute c = 7 and b = 8 into the expression for a:
a = -9 + c
a = -9 + 7
a = -2
Therefore, the values of a, b, and c that satisfy all three equations are a = -2, b = 8, and c = 7.
To find the value of a+b+c, simply add the three values together:
a + b + c = -2 + 8 + 7 = 13
The value of a+b+c is 13.