Problem 19 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-1,\) passing through \(\left(-\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(x + \frac{1}{2} = - (y + 2)\) and in slope-intercept form it is \(y = -x - \frac{5}{2}\).

Step 1: Write the Equation in Point-Slope Form

With slope \(m = -1\), and coordinates \(\left(-\frac{1}{2},-2\right)\), point-slope form is given as \(y - y_1 = m(x - x_1)\). Substituting the given values, we get: \(y - (-2) = -1 (x - -\frac{1}{2})\). Simplify this equation for a final point-slope form.

Step 2: Simplify the Point-Slope Equation

Simplifying the equation \(y - (-2) = -1 (x - -\frac{1}{2})\), gives us \(y + 2 = -(x + \frac{1}{2})\). For further simplification, we remove the negative sign by swapping the sides, to get \(x + \frac{1}{2} = - (y + 2)\). This is the final point-slope form.

Step 3: Write the Equation in Slope-Intercept Form

To write the equation in slope-intercept form \(y = mx + c\), we have to bring \(y\) alone on one side. Using the equation from Step 2, \(x + \frac{1}{2} = - (y + 2)\), we isolate \(y\) to get the final slope-intercept form.

Step 4: Simplify the Slope-Intercept Equation

By simplifying the equation \(x + \frac{1}{2} = - (y + 2)\), we get \(y = -x - \frac{5}{2}\). This is the final slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is an equation of a straight line and easily reveals both the slope and the y-intercept of the line. It has a general format of \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) indicates the y-intercept, which is the point where the line intersects the y-axis.

Understanding slope-intercept form is crucial for graphing linear equations efficiently. For the student exercise where the slope is given as \( -1 \), and the line passes through the point \( \left(-\frac{1}{2},-2\right) \), we can rearrange the point-slope form equation into the slope-intercept form. The equation transitions from \( y + 2 = -(x + \frac{1}{2}) \) after isolation and simplification to \( y = -x - \frac{5}{2} \), where \( -1 \) is the slope (\( m \)), and \( -\frac{5}{2} \) is the y-intercept (\( b \)).

Students sometimes struggle with this form when attempting to identify the y-intercept directly from the equation. It's fundamental to remember that the y-intercept is the part of the equation without the variable \( x \), alone on one side of the equation.

Linear Equations

Linear equations are the foundation for exploring relationships between two variables in algebra. They are equations where any variables appear to the first degree, meaning they are not squared, cubed, or taken to any higher power. The equation of a line in two dimensions is always a linear equation.

The exercise provided is a classic example of such a linear relationship. Starting with point-slope form and transitioning to slope-intercept highlights this linearity. In the general form of a linear equation, \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, adjusting it into the slope-intercept form makes it visually clear that changes in \( x \) produce straight-line changes in \( y \).

Tips for Dealing with Linear Equations

• Always try to simplify the equation as much as possible.
• Rearrange the equation to isolate the variable of interest if necessary.
• Check your work by substituting points to ensure they satisfy the equation.

For better understanding, practice graphing the linear equation to visualize how changing parameters affect the line's position and direction.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For example, in the exercise, we encounter expression on each side of the equation, like \( -(x + \frac{1}{2}) \) and \( y + 2 \), before reaching the more simplified slope-intercept form.

It's important to distinguish algebraic expressions from equations. An expression cannot be 'solved' like an equation because it does not have an equality sign; it's a phrase, not a full sentence. When we manipulate algebraic expressions, our goal is to simplify them or transform them into a desired form.

There are a few steps students can take to manage algebraic expressions effectively:

• Combine like terms wherever possible to simplify expressions.
• Apply distributive properties to remove parentheses.
• Be careful with signs, especially when working with negative signs and subtraction.

By understanding and applying these principles, students can work through the algebraic expressions embedded within the slope-intercept form of the equation.