Problem 11 - 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. Slope \(=\frac{1}{2},\) passing through the origin

Step-by-Step Solution

Answer

The point-slope form of the equation of the line is \(y = \frac{1}{2}x\) and the slope-intercept form is also \(y = \frac{1}{2}x\).

Step 1: Determine the Point-Slope Form

The point-slope form of an equation is \(y - y_1 = m(x - x_1)\). We know that the slope \(m = \frac{1}{2}\) and the line passes through the origin which is the point \((0,0)\), so \(x_1 = 0\) and \(y_1 = 0\). Substituting these values in the formula, we get \(y - 0 = \frac{1}{2} (x - 0)\), which simplifies to \(y = \frac{1}{2}x\).

Step 2: Determine the Slope-Intercept Form

The slope-intercept form of an equation is \(y = mx + b\). Here, we already found that \(m = \frac{1}{2}\) and the y-intercept \(b\) is the y-coordinate of the point where the line intersects the y-axis. Since the line passes through the origin, the y-intercept is also 0. Thus \(y = \frac{1}{2}x + 0\), simplifying to \(y = \frac{1}{2}x\). Hence, the slope-intercept form is the same as the point-slope form in this case.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is a fundamental skill in algebra that helps to easily graph and analyze linear equations. The generic slope-intercept form is represented by the equation \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept — the point at which the line crosses the y-axis.

In the context of the textbook exercise, the slope-intercept form simplifies to \(y = \frac{1}{2}x\), because the y-intercept (\(b\)) is 0. This line is a straight line that increases at a constant rate, and for every one unit that x increases, y increases by half a unit.

This form is particularly helpful when graphing because it gives you the start point (the y-intercept) and the direction and steepness of the line (the slope). Knowing how to convert from point-slope to slope-intercept form, as shown in the solved exercise, allows students to easily plot linear equations on a graph.

Linear Equations

Linear equations create the straight lines you see on a graph. They are called 'linear' because the highest power of the variable is 1, leading to a line with constant slope. The exercise provided demonstrates a linear equation in point-slope form transitioning to slope-intercept form.

Understanding linear equations is crucial for solving a variety of real-world problems. They can represent relationships between distance and time, cost and quantity, and many other pairs of measures. In the example, the equation \(y = \frac{1}{2}x\) can represent a situation where y could be the total cost, and x the number of items purchased, with each item costing half a unit of currency.

Tips for Understanding Linear Equations:

• Identify the slope and y-intercept from an equation in slope-intercept form.
• Plot the y-intercept on the graph first, then use the slope to find another point on the line.
• Remember that the slope is a ratio that represents how y changes with x.

Equations like these are building blocks for more advanced mathematical concepts and are widely applicable in many fields, including science, business, and economics.

Coordinates

Coordinates are a set of values that show an exact position on a graph or a map. In a two-dimensional space, coordinates are represented by a pair of numbers \(x,y\), which correspond to the horizontal and vertical positions, respectively. For the exercise given, the coordinates of the origin are \(0,0\), which means it lies at the point where the x-axis and y-axis intersect.

When you're learning about points on a graph, remember:

• The first number in the pair, known as the x-coordinate, shows how far along to move from the origin.
• The second number, the y-coordinate, shows how far up or down to move from the origin.

Knowing how to read and plot coordinates is essential for creating graphs and interpreting data. In linear equations, using coordinates to identify specific points, like the y-intercept, allows for an accurate representation of the relationship between variables on a graph.

Thus, in the exercise's context, identifying the origin as the point through which the line passes, and understanding that the origin's coordinates are \(0,0\) was crucial to simplifying the equation into slope-intercept form.