Add or subtract
[tex]$\frac{7-x}{x-3}-\frac{2 x-5}{3-x}$[/tex]
A. [tex]$-\frac{x+2}{x-3}$[/tex]
B. [tex]$\frac{x+2}{x-3}$[/tex]
C. [tex]$\frac{x+12}{x-3}$[/tex]
D. [tex]$-\frac{x+12}{x-3}$[/tex]
Answer (1)
Rewrite the expression with a common denominator of x − 3 .
Combine the numerators: ( 7 − x ) + ( 2 x − 5 ) − ( x + 2 ) + ( x + 2 ) − ( x + 12 ) .
Simplify the numerator to − 10 .
The simplified expression is x − 3 − 10 .
Explanation
Understanding the Expression We are given the expression x − 3 7 − x − 3 − x 2 x − 5 − x − 3 x + 2 + x − 3 x + 2 − x − 3 x + 12 and we want to simplify it.
Rewriting with Common Denominator Notice that 3 − x = − ( x − 3 ) . We can rewrite the second term to have the same denominator as the others: x − 3 7 − x − 3 − x 2 x − 5 = x − 3 7 − x + x − 3 2 x − 5 Now the expression becomes x − 3 7 − x + x − 3 2 x − 5 − x − 3 x + 2 + x − 3 x + 2 − x − 3 x + 12 Since all terms now have the same denominator, we can combine them into a single fraction:
Combining the Fractions x − 3 ( 7 − x ) + ( 2 x − 5 ) − ( x + 2 ) + ( x + 2 ) − ( x + 12 ) Now, we simplify the numerator by combining like terms:
Simplifying the Numerator 7 − x + 2 x − 5 − x − 2 + x + 2 − x − 12 = ( − x + 2 x − x + x − x ) + ( 7 − 5 − 2 + 2 − 12 ) = 0 x − 10 = − 10 So the expression simplifies to:
Final Result x − 3 − 10 Therefore, the simplified expression is x − 3 − 10 .
Examples
This type of algebraic simplification is used in many areas of engineering and physics, such as when analyzing circuits or mechanical systems. For example, when dealing with electrical circuits, you might need to combine several impedance terms (which can be complex fractions) to find the total impedance of the circuit. Simplifying these expressions allows engineers to more easily analyze the behavior of the circuit and design it effectively. Similarly, in mechanical systems, simplifying expressions involving forces and moments can help in understanding the stability and response of the system.
