Solving a System of Equations: A Step-by-Step Guide
With a bit of algebraic manipulation and substitution, we can solve complex systems of equations. This article will walk you through solving the following system of equations:
1. Problem Setting
We are given two equations:
7a 4b c 51 Equation 1 3a 4b 5c 15 Equation 2Our task is to find the value of abc.
2. Mathematical Approach
To solve for abc, we will eliminate one of the variables by manipulating the equations. Let's follow the steps:
2.1 Isolate #949; in Equation 1
c 51 - 7a - 4b
2.2 Substitute this expression for #949; into Equation 2
3a 4b 5(51 - 7a - 4b) 15
Expanding this gives:
3a 4b 255 - 35a - 20b 15
2.3 Combine Like Terms
3a - 35a 4b - 20b 255 15
Which simplifies to:
-32a - 16b 255 15
2.4 Isolate the Variable Terms
-32a - 16b 15 - 255
Which further simplifies to:
-32a - 16b -240
2.5 Divide the Entire Equation by -16 to Simplify
2a b 15 text{ Equation 3}
2.6 Express b in Terms of a
b 15 - 2aquadtext{Equation 4}
2.7 Substitute Equation 4 into Equation 1
7a 4(15 - 2a) c 51
Expanding gives:
7a 60 - 8a c 51
Combining like terms:
-a 60 c 51
2.8 Isolate c
c 51 - a 60
Which simplifies to:
c a - 9quadtext{Equation 5}
2.9 Find a, b, and c
b 15 - 2a c a - 92.10 Calculate abc
acdot bcdot c acdot (15 - 2a) cdot (a - 9)
Combining terms:
acdot (15 - 2a) cdot (a - 9) 15 - 9 6
3. Conclusion
The value of abc is:
boxed{6}