91Ó°ÊÓ

Substitution is a fundamental concept in algebra. It involves replacing variables in an expression with specific values. In this exercise, we are given the values:

a = -1
b = 3
c = -4

We substitute these values into the expression \(\frac{b-c}{b-a}\).
This step is crucial because it converts an abstract algebraic expression into a specific numerical one, making it easier to evaluate. After substitution, our expression becomes: \(\frac{3-(-4)}{3-(-1)}\).

Remember, when substituting values, pay close attention to signs (positive or negative) between the variables and numbers.

Simplification is the process of making an expression easier to understand or solve. After substitution, we must simplify both the numerator and the denominator separately.

1. **Numerator Simplification:**
The numerator is \(3 - (-4)\), which simplifies to \(3 + 4 = 7\) because subtracting a negative number is the same as adding its positive counterpart.

2. **Denominator Simplification:**
The denominator is \(3 - (-1)\), which simplifies to \(3 + 1 = 4\) for the same reason.

These steps transform a complicated fraction into a simpler fraction that is easy to evaluate.

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. In this exercise, our expression \(\frac{b-c}{b-a}\) is an example of an algebraic fraction.

After substitution and simplification, we get the simplified fraction \(\frac{7}{4}\). The final step is to evaluate this algebraic fraction by performing the division: \[ \frac{7}{4} = 1.75 \].

Understanding how to handle algebraic fractions is important because they frequently appear in algebra problems. When dealing with algebraic fractions, always handle the numerator and denominator separately at first. Then, simplify the fraction to make the problem more manageable.

In summary, by substituting specific values for the variables and carefully simplifying the expression, we transformed a complex algebraic fraction into a simple numerical value.