Problem 17 - Thomas Calculus

Question

In Exercises 17–30, write an equation for each line described. Passes through \((-1,1)\) with slope \(-1\)

Step-by-Step Solution

Answer

The equation is \( y = -x \).

Step 1: Review the Point-Slope Formula

The point-slope form of a line's equation is given by:\[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. In this problem, we have the point \((-1, 1)\) and the slope \(-1\).

Step 2: Substitute the Given Values into the Formula

Substitute \(x_1 = -1\), \(y_1 = 1\), and \(m = -1\) into the point-slope formula:\[ y - 1 = -1(x + 1) \]

Step 3: Simplify the Equation

Distribute the slope \(-1\) on the right-hand side of the equation:\[ y - 1 = -1x - 1 \]

Step 4: Solve for y

Add 1 to both sides to solve for \(y\):\[ y = -1x \] or simply \[ y = -x \]

Step 5: Final Equation

The equation of the line that passes through \((-1, 1)\) with a slope of \(-1\) is:\[ y = -x \]

Key Concepts

Linear Equations

Linear equations represent straight lines on a graph. When we talk about linear equations in algebra, we mean equations that take the form of expressions with variables raised to the power of one. These equations are called "linear" because their graph is always a straight line.
In a 2D coordinate plane, a common example of a linear equation is the formula for a line, which can be expressed in different forms, such as point-slope form or slope-intercept form.

• Help visualize data relationships through straight lines.
• Can be transformed between different forms like point-slope, slope-intercept, etc.

Linear equations have various real-world applications such as in predicting trends, calculating distances, or even budgeting expenses. Once the equation is formulated, its solutions can usually be easily found through simple algebraic manipulation.

Equation of a Line

The equation of a line is the mathematical statement that describes the line's properties, such as angle and where it crosses the axes. The point-slope form and slope-intercept form are just two of the most common ways to write this equation.
This exercise demonstrates how to use the point-slope form of the line equation to derive the line's formula that fits specific data:

• Point-Slope Form: Given by the equation \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.
• Finding a line equation requires substituting known values into one of these forms.

No matter how you write an equation of a line, its core purpose is to define a linear trajectory on a 2D plane.

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to represent the equation of a line. It is given by the formula:\[ y = mx + b \]where \(m\) represents the slope, and \(b\) represents the y-intercept where the line crosses the y-axis.
This format is especially useful in quickly identifying two primary characteristics of the line:

• The slope \(m\), indicating the steepness or incline of the line.
• The y-intercept \(b\), showing where the line meets the y-axis.

In the case of our exercise, after substituting values from the point-slope form, we simplified it to the slope-intercept form as \( y = -x \), which beautifully displays both the slope (-1) and the y-intercept (0). This make it very easy to plot on the graph.