Problem 7 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((0,6)\) with slope 5

Step-by-Step Solution

Answer

y = 5x + 6

Step 1: Identify the Slope and Point

We are given a point \( (0,6) \) and a slope of 5.

Step 2: Use the Point-Slope Form

The point-slope form of a linear equation is given by \[ y - y_1 = m(x - x_1) \], where \( m \) is the slope and \( (x_1, y_1) \) is the point.

Step 3: Substitute the Given Values

Substitute the slope 5 and the point \( (0, 6) \) into the point-slope form: \[ y - 6 = 5(x - 0) \].

Step 4: Simplify the Equation

Simplify the equation to get it into the slope-intercept form \( y = mx + b \): \[ y - 6 = 5x \]. Add 6 to both sides: \[ y = 5x + 6 \].

Step 5: Final Equation

The equation of the line is \( y = 5x + 6 \).

Key Concepts

Point-Slope Form

Understanding the point-slope form of a linear equation can be very helpful when you know a specific point and the slope of a line. The general formula is given as: \[ y - y_1 = m(x - x_1) \]where:

• m denotes the slope of the line
• (x_1, y_1) represents the coordinates of a known point on the line

The left side of the equation, y - y1, represents the vertical change from the known point, while the right side, m(x - x1), represents the horizontal change, multiplied by the slope (m). Suppose you're given a point (0, 6) and a slope of 5. Plugging in these values into the point-slope form, you get \[ y - 6 = 5(x - 0) \] This equation captures the relationship between x and y for all points on the line.

Slope-Intercept Form

The slope-intercept form of a linear equation is another way to express the equation of a line. The formula is \[ y = mx + b \], where

• m is the slope
• b is the y-intercept (the value of y when x is 0)

This form is useful for quickly identifying the slope and the y-intercept of the line. Once you have your equation in the point-slope form, converting it to the slope-intercept form is straightforward. From our previous example, we have \[ y - 6 = 5(x - 0) \] Simplify it to get \[ y = 5x + 6 \] Here, the slope (m) is 5, and the y-intercept (b) is 6. You can now easily graph this line by starting at the y-intercept (6) and using the slope (rise over run) to find another point.

Equation of a Line

The equation of a line represents all the points that lie on that line. Various forms of linear equations suit different purposes. We've discussed the point-slope and slope-intercept forms, but remember that there are others, such as the standard form \[ Ax + By = C \] Regardless of the form, they can be transformed into one another. To recap:

• The point-slope form is good for when you know one point and the slope
• The slope-intercept form is great for quickly identifying the slope and y-intercept

Recognizing which form to use can simplify your work. For instance, if you need to determine the equation of a line passing through (0, 6) with a slope of 5, you can write it as \[ y - 6 = 5(x - 0) \] (point-slope) and simplify to \[ y = 5x + 6 \] (slope-intercept). These are two ways to describe the same line, helping you better understand and graph them.