My teacher said we are supossed to solve this using systems of equations:
Let a, b, and c be real numbers such that a-7b+8c=4 and 8a+4b-c=7. Then a^2-b^2+c^2= ?
can someone please help?! i don't know what method to use to solve this or where to start
What you'd like to do is to solve the simultaneous equations
a-7b+8c=4
8a+4b-c=7
for a and b in terms of c.
Substitute the values of a and b in the expression a²+b²+c² and wait for the magic to happen.
If you do your math correctly, magic will happen.
To solve this problem using systems of equations, we first need to eliminate one variable by adding or subtracting the equations. Let's start by multiplying the second equation by -1:
-(8a + 4b - c) = -(7)
This gives us:
-8a - 4b + c = -7
Now we have two equations:
a - 7b + 8c = 4 ...(1)
-8a - 4b + c = -7 ...(2)
Next, we can add these two equations together to eliminate the "c" variable:
(a - 7b + 8c) + (-8a - 4b + c) = 4 - 7
This simplifies to:
-7a - 11b = -3 ...(3)
Now we have two equations:
-7a - 11b = -3 ...(3)
a - 7b + 8c = 4 ...(1)
Now, we have a system of two equations with two variables. We can use any method to solve this system. For this example, let's use the substitution method:
From equation (3), we can solve for "a" in terms of "b":
a = (-3 + 11b) / -7
Now substitute this expression for "a" into equation (1):
((-3 + 11b) / -7) - 7b + 8c = 4
Simplify the equation by multiplying through by -7 to eliminate the fraction:
-3 + 11b + 49b - 56c = -28
Combine like terms:
60b - 56c = -25 ...(4)
Now we have two equations:
60b - 56c = -25 ...(4)
a - 7b + 8c = 4 ...(1)
To find a numerical value for a^2 - b^2 + c^2, we need to solve for the values of a, b, and c in this system. To do this, we can use any suitable method, such as substitution or elimination.
Let's use the elimination method to solve for c:
Multiply equation (1) by 8:
8a - 56b + 64c = 32 ...(5)
Now add equations (4) and (5) to eliminate c:
(60b - 56c) + (8a - 56b + 64c) = -25 + 32
Simplify the equation:
8a + 4b = 7 ...(6)
Now we have two equations:
8a + 4b = 7 ...(6)
60b - 56c = -25 ...(4)
We can choose to solve either equations (6) or (4) for one variable and substitute it back into the other equation. Let's solve equation (6) for a:
8a = 7 - 4b
a = (7 - 4b) / 8
Substitute this expression for "a" into equation (4):
60b - 56c = -25
Simplify the equation:
(60b - (56((7 - 4b) / 8))) = -25
Solve for c:
60b - (7 - 4b) = -25
Combine like terms:
64b - 7 = -25
64b = -18
b = -18 / 64 = -9 / 32
Substitute the value of b back into the expression for a:
a = (7 - 4b) / 8 = (7 - 4(-9 / 32)) / 8 = (7 + 36 / 32) / 8 = (56 + 36) / 32 = 92 / 32 = 23 / 8
Similarly, we substitute the values of a and b back into equation (1) to solve for c:
a - 7b + 8c = 4
(23 / 8) - 7(-9 / 32) + 8c = 4
(23 / 8) + (63 / 32) + 8c = 4
(23 / 8) + (63 / 32) + 8c = 4
Multiply through by the common denominator (32):
(184 + 63 + 256c) / 32 = 128 / 32
Combine like terms:
(247 + 256c) / 32 = 128 / 32
To make this equation easier to solve, we can multiply both sides by 32 to eliminate the fraction:
247 + 256c = 128
Subtract 247 from both sides:
256c = -119
Divide by 256 to solve for c:
c = -119 / 256
Finally, substitute the values of a, b, and c into the expression a^2 - b^2 + c^2:
(a^2) - (b^2) + (c^2) = (23/8)^2 - (-9/32)^2 + (-119/256)^2
Simplifying this expression will give you the final answer.