Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-3,0)\) and \((0,3)\)

Step-by-Step Solution

Answer

The point-slope form of the equation is \(y=x+3\) and the slope-intercept form is also \(y=x+3\).

Step 1: Find slope

Initially we need to find the slope of the line passing through the two points. The formula for the slope is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Substituting the given points \((-3,0)\) and \((0,3)\) into this equation gives us \(m=\frac{3-0}{0-(-3)}=1\).

Step 2: Point-slope form

Next we substitute the slope \(m\) and the point \((-3,0)\) into the point-slope formula: \(y-y_1=m(x-x_1)\). This gives us \(y-0=1(x-(-3))\) or \(y=x+3\). This is the point-slope form of the equation.

Step 3: Slope-intercept form

The point-slope form of the equation we got is already in slope-intercept form \(y=mx+b\) since it is expressed as \(y=x+3\). Therefore, there are no steps required to transform it further.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used forms when dealing with straight lines in coordinate geometry. It is expressed as
\( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis. Understanding this form is crucial in algebra as it allows you to quickly graph a line and comprehend its behavior by just looking at these two coefficients. For example, given the equation \( y = x + 3 \), we can see that the slope is 1, and the y-intercept is 3. This indicates a line rising diagonally upward from the left to the right and crossing the y-axis at the point (0,3).

Slope of a Line

The slope of a line is a measure of its steepness and is calculated as the ratio of the rise (the change in y) over the run (the change in x). Algebraically, it is represented as
\( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \).
Knowing the slope is vital as it tells us how fast y is changing with respect to x. A positive slope means the line is ascending, a negative slope indicates a descending line, and a slope of zero signifies a horizontal line. In our example with points \( (-3,0) \) and \( (0,3) \), we find the slope by substituting the coordinates into the slope formula, yielding a slope of 1. This tells us the line goes up one unit for every unit it goes right.

Algebraic Equations

Algebraic equations are the backbone of algebra and represent the relationship between variables using mathematical operations. They can range from simple, linear equations to more complex ones involving exponents and variables. The process of solving an equation often involves isolating the variable on one side to find its value. In the context of linear equations and coordinate geometry, we deal with equations representing lines. By converting equations to different forms, such as point-slope or slope-intercept form, we can easily interpret the characteristics of the line, such as direction, steepness, and points of intersection with the axes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This branch of mathematics allows us to represent geometric shapes algebraically and solve geometric problems using algebraic techniques. Key elements in coordinate geometry include points, lines, and curves, each represented by algebraic equations or inequalities. The Cartesian coordinate system, with its perpendicular x and y-axes, is especially useful for representing linear equations and analyzing their properties. For instance, by plotting points \( (-3,0) \) and \( (0,3) \) on a graph and connecting them, we visualize the line and can confirm the slope and intercepts we calculated algebraically.