Problem 23 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-1,0)\) and \((0,-1)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -x - 1 \).

Step 1: Calculate the slope

First, find the slope of the line passing through the points \((-1, 0)\) and \((0, -1)\). Use the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points \( (x_1, y_1) = (-1, 0) \) and \( (x_2, y_2) = (0, -1) \):\[ m = \frac{-1 - 0}{0 + 1} = \frac{-1}{1} = -1 \]

Step 2: Use the point-slope form

Next, use the point-slope form of the equation of a line, which is given by:\[ y - y_1 = m(x - x_1) \]Substitute one of the points \( (x_1, y_1) = (-1, 0) \) and the slope \(m = -1\):\[ y - 0 = -1(x + 1) \]Simplify this equation:\[ y = -x - 1 \]

Step 3: Convert to slope-intercept form

Since the simplified equation is already in slope-intercept form \( y = mx + b \), the equation of the line is:\[ y = -x - 1 \]

Key Concepts

Slope Formula

When determining the equation of a line that passes through two given points, the first step is to find the slope.

The slope formula is a method used to calculate the steepness or inclination of a line. This formula is essential for understanding how much the y-coordinate changes as the x-coordinate changes.Use this formula to find the slope (m):
\text\text\text\text\text\text\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
You substitute the coordinates of the two points (\text\text (x_1, y_1)\text\text\text\text and \text\text (x_2, y_2)) into this formula.

• For the points (-1, 0) and (0, -1), your coordinates are (x_1, y_1) = (-1, 0) and (x_2, y_2) = (0, -1).
• After substituting these into the formula, you calculate:
  \text\text\[ m = \frac{-1 - 0}{0 + 1} = \frac{-1}{1} = -1 \]
  
  

So, the slope m is -1.

Point-Slope Form

Once the slope is known, the next step is to use the point-slope form of the line's equation. The point-slope form is handy when you have one point over the line and the slope.
Point-slope form equation:
\text\text\text\text\text\text\text\[ y - y_1 = m(x - x_1) \]

• Here, \text\text(x_1, y_1) is the point(-1, 0), and m is your slope which is -1.
Use the point-slope form and substitute them in the formula:

After substitution, the equation becomes: \text\text\[ y - 0 = -1(x + 1) \]

all simplified yields:
\[ y = -x - 1 \] Thus, you have found equation required: \text \text\[ y = -x -1 \]

Slope-Intercept Form

Slope-intercept form of an equation is another way to represent a line. It is often considered the most straightforward form since it explicitly shows the slope and the y-intercept.
Slope-intercept form equation:
\text\text\[ y = mx + b \]

• Here \(m) is the slope and \)b) is the y-intercept,y-coordinate of the point where line intersects.

let's convert our point-slope eq<\(y - y_1 = m(x -x_1), \)

We substitute each variable with appropriate values:
(x1,y1)= (-1,0) along with slope(m=1) which results =\-(y - 0 = x +1)practical reconsideration,

We get finally of:-(m) place,smtht in,intercept form\[ y= -x-1\]