Problem 13 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=6,\) passing through (-2,5)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y - 5 = 6 (x + 2)\) and in slope-intercept form is \(y = 6x + 17\).

Step 1: Write the Point-Slope Form of Equation

We begin with recalling the point-slope form equation using the given point and slope. The equation of a line in point-slope form when a point \((x_1,y_1)\) and slope \(m\) are given is: \(y - y_1 = m (x - x_1)\). Substituting the given values, i.e., \(x_1 = -2, y_1 = 5, m = 6\), yields the equation in point-slope form as \(y - 5 = 6 (x + 2)\).

Step 2: Write the Slope-Intercept Form of Equation

Knowing the equation in the point-slope form, it is possible to convert it into slope-intercept form. The slope-intercept form is given as \(y = mx + c\). Here, expanding the equation \(y - 5 = 6 (x + 2)\) gives \(y = 6x + 12 + 5\). This simplifies to \(y = 6x + 17\).

Step 3: Present the Result

After computation, we have the equations of the line that has a slope 6 and passes through the point (-2,5). In point-slope form, it is \(y - 5 = 6 (x + 2)\), and in slope-intercept form, it is \(y = 6x +17\).

Key Concepts

Slope-Intercept Form Equation

Understanding the slope-intercept form of a linear equation is essential for graphing and analyzing lines. This special form is expressed as

\( y = mx + b \)

where \( m \) represents the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis. The slope, \( m \), indicates the steepness of the line—how much the line rises or falls as it moves from left to right. The y-intercept, \( b \), gives us a starting point to graph the line.

By simply knowing the slope and y-intercept, we can quickly sketch the line without plotting multiple points. To convert the point-slope form to the slope-intercept form, isolate \( y \) on one side of the equation.

The process usually involves expanding the equation if it is in the format of point-slope form and then simplifying it. For instance, if we have a line with a slope of 6 that passes through the point (-2,5), as seen in the exercise, the equation starts in point-slope form as:

\( y - 5 = 6(x + 2) \)

After expanding and simplifying, we have:

\( y = 6x + 17 \)

showing that the line has a slope of 6 and a y-intercept at \( (0, 17) \). This makes plotting the line simple and straightforward.

Linear Equations

Linear equations form the basis for a wide range of mathematical concepts and real-world applications. They create straight lines when graphed on a coordinate plane and have the general form of:

\( ax + by = c \)

where \( a \), \( b \), and \( c \) are constants. These equations showcase a relationship where the dependent variable, \( y \), changes at a constant rate with respect to the independent variable, \( x \). This rate of change is known as the slope.

The beauty of linear equations lies in their simplicity and the ease with which they can be manipulated into different forms—such as the slope-intercept form or the point-slope form—to suit various purposes, from solving systems of equations to modeling real-life situations. A fundamental skill in working with linear equations is being able to identify the slope and y-intercept, which allow us to write the equation in its slope-intercept form as discussed earlier.

Writing Equations of Lines

Writing the equation of a line is a key skill in algebra that allows us to model and solve problems involving linear relationships. There are several ways to write the equation of a line, depending on the given information:

• If the slope \( m \) and y-intercept \( b \) are known, the slope-intercept form \( y = mx + b \) can be used.
• When given a slope and a point, or two points, the point-slope form, \( y - y_1 = m(x - x_1) \), is often the easiest approach.
• If the x-intercept and y-intercept are known, using the intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \) might be preferred.

Using the point-slope form is particularly helpful when we're given the slope of a line and a specific point it passes through. As illustrated in the exercise, we started with a slope of 6 and the point (-2, 5) to write the point-slope equation. From there, we were able to convert it to slope-intercept form, a versatile and commonly used representation of a linear equation.

These various forms are tools in the algebra toolkit, and being fluent in using them allows students to approach problems with flexibility, selecting the most efficient method for the situation at hand.