Simplifying rational algebraic expressions, how to plot points in the rectangular coordinate system, linear equations in two variables, slope of a line.

Steps to Solve

1. Identify the Rational Expression Start with the rational expression that needs simplification. For instance, assume we have $\frac{x^2 - 1}{x - 1}$.

2. Factor the Numerator Factor the numerator to simplify the expression. In our example: $$x^2 - 1 = (x - 1)(x + 1)$$ So, we rewrite the expression as: $$\frac{(x - 1)(x + 1)}{x - 1}$$

3. Cancel Common Factors Next, cancel out the common factors in the numerator and denominator: $$\frac{(x - 1)(x + 1)}{x - 1} = x + 1, \text{ for } x \neq 1$$

4. Plotting Points in a Rectangular Coordinate System If you need to plot the simplified expression, you identify points. For each value of $x$, calculate the corresponding $y$ value (from $y = x + 1$). For example:

• For $x = 0$, $y = 0 + 1 = 1$ (point (0, 1))
• For $x = 1$, $y = 1 + 1 = 2$ (point (1, 2))
• For $x = -1$, $y = -1 + 1 = 0$ (point (-1, 0))

5. Understanding Linear Equations Recognize that the expression $y = x + 1$ is a linear equation in two variables. It shows a straight line with a slope.

6. Finding the Slope of the Line The slope of the line represented by $y = mx + b$ is $m$, where $m$ is the coefficient of $x$. Here, $m = 1$, which means the slope is $1$.