Problem 46 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of line that passes through the given points. \(m=\frac{1}{2}, b=0\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that has a slope of \(1/2\) and passes through the y-axis at the origin (0,0) is \(y = (1/2)x\).

Step 1: Identify the given slope and y-intercept

The slope \(m\) is given as 1/2 and the y-intercept \(b\) is given as 0.

Step 2: Write down the general equation of a line in slope-intercept form

The general form of the equation of a line in slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Plug the values of slope and y-intercept into the general equation of a line

Substituting the provided values into the equation yields \(y = (1/2)x + 0\).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental to algebra and are the representation of lines on a graph. In essence, a linear equation provides a relationship between two variables, usually x and y, where the graph of these equations is always a straight line. This straight line can have various slopes and positions, which are determined by the specific values within the equation.

The standard form of a linear equation is expressed as ax + by = c, where a, b, and c are constants. However, for many practical applications, such as predicting outcomes and solving real-world problems, the slope-intercept form, y = mx + b, is preferred due to its direct display of the slope and y-intercept. Understanding linear equations is crucial as they are used to model and solve problems involving rates of change and trends.

Exploring Slope and Y-intercept

When it comes to linear equations, two critical components are the slope and the y-intercept. The slope, denoted by the letter m, measures the steepness or inclination of a line. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line, often represented as \( m = \frac{rise}{run} \).

The y-intercept, represented by the letter b, is the point where the line crosses the y-axis. This is the value of y when x is zero. Together, the slope and y-intercept define the unique position and direction of a line on a Cartesian plane.

For example, in the equation \( y = mx + b \), a slope of \(\frac{1}{2}\) implies that for every one unit of horizontal movement (to the right for positive, to the left for negative), the line rises half a unit. If the y-intercept is 0, the line crosses the origin of the graph, which also emphasizes the line's proportionality.

Writing Equations of Lines

Writing the equation of a line involves understanding its slope (m) and y-intercept (b). The slope-intercept form, \( y = mx + b \), is particularly useful for quickly sketching the graph of a line or deducing its characteristics.

When you're given the slope and y-intercept directly, as in the exercise with \(m=\frac{1}{2}, b=0\), writing the equation is straightforward. You simply plug these values into the slope-intercept formula to obtain \(y = \left(\frac{1}{2}\right)x + 0\), which simplifies to \(y = \frac{1}{2}x\).

Visualizing the Equation

One of the best ways to understand the equation is by plotting it on a graph. With our given equation of \(y = \frac{1}{2}x\), you will notice that the line passes through the origin and for each step right on the x-axis (positive direction), you go up half a step on the y-axis. This visualization confirms that the slope is 1/2 and that the y-intercept is indeed 0, as the line crosses the y-axis at the origin.