Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(57-64\), write an equation in the form \(y=m x+b\) of the line that is described. The line rises from left to right. It passes through the origin and a second point with equal \(x\) - and \(y\) -coordinates.

Step-by-Step Solution

Answer

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

Step 1: Determine the slope (m)

The slope of a line is the change in \(y\) divided by the change in \(x\) (also known as 'rise over run'). We know the line passes through the origin, meaning we have one point which is (0,0), and another where the \(x\) and \(y\) coordinates are equal, meaning the second point is (1,1). The slope \(m\) is then calculated as \(m = (y_2 - y_1) / (x_2 - x_1)\). Using our points, this becomes \(m = (1 - 0) / (1 - 0) = 1\).

Step 2: Determine the y-intercept (b)

Since the line passes through the origin (0,0), the y-intercept (the point where the line crosses the y-axis) is 0.

Step 3: Write the equation in slope-intercept form

Now that we have the slope and y-intercept, the equation of the line in slope-intercept form \(y=mx+b\) is \(y = 1*x + 0\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to represent a linear equation. It is given by the formula \(y = mx + b\). In this form:

• \(y\) represents the dependent variable or the output value.
• \(m\) represents the slope of the line, describing how steep the line is.
• \(x\) is the independent variable or the input value.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

The slope indicates how quickly \(y\) changes with respect to \(x\). A positive slope means the line rises as \(x\) increases. A zero slope implies a horizontal line, whereas a negative slope indicates the line falls as \(x\) increases.
Writing an equation such as \(y = x\) in slope-intercept form simplifies the understanding of linear relationships. In this case, the slope \(m\) is 1, and the y-intercept \(b\) is 0. This means the line passes through the origin and is a perfect diagonal where \(y\) increases exactly as \(x\) does.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses a coordinate plane to define and describe geometric shapes. A coordinate plane consists of an x-axis (horizontal) and a y-axis (vertical), intersecting at the origin point (0, 0).
Every point on the plane is described by a pair of numbers \((x, y)\). These coordinates specify the location relative to the axes. The concept extends to lines, curves, and other shapes by plotting points that describe their position and characteristics.

• Lines are determined by their slope and a point or by an equation in specific forms like the slope-intercept form \(y = mx + b\).
• Shapes such as circles, rectangles, and triangles can also be represented with equations derived from their geometric properties and positions.

By understanding the coordinate plane and the relationships between different points and lines, one can solve a variety of geometric problems using algebraic methods, making it a powerful tool in mathematics.

Origin Point

The origin point in a coordinate plane is the point where the x-axis and y-axis intersect. It is denoted by the coordinates (0, 0).
The origin is a fundamental reference point in coordinate geometry.

• It serves as the starting point for defining other points in the plane using coordinates.
• It simplifies many equations because calculations often begin from this zero point.
• In the slope-intercept form equation, if a line passes through the origin, the y-intercept \(b\) is zero.

For example, a line described by the equation \(y = x\) passes through the origin, illustrating a direct proportional relationship between \(x\) and \(y\). Understanding the origin helps visualize and draft lines and shapes more accurately on the coordinate plane. Knowing that the origin is always (0, 0) allows for straightforward calculations and simplifications in geometry and algebraic expressions.