Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Slope \(=\frac{1}{2} ;\) containing the point (3,1)

Short answer

The equation of the line is \( y = \frac{1}{2}x - \frac{1}{2}\).

Step-by-step solution

- Identify the slope and point

The slope of the line is given as \(\frac{1}{2}\) and it contains the point (3,1). This point is denoted as (x₁, y₁) = (3, 1).

- Use the point-slope formula

The point-slope formula for a line is \(y - y₁ = m(x - x₁)\), where \(m\) is the slope and \( (x₁, y₁) \) is a point on the line. Substituting the given values, we get: \(y - 1 = \frac{1}{2}(x - 3)\).

- Simplify the equation

Now, distribute the slope and simplify the equation: \(y - 1 = \frac{1}{2}x - \frac{3}{2}\). To solve for \(y\), add 1 to both sides: \(\begin{aligned}y - 1 & = \frac{1}{2}x - \frac{3}{2} \ y & = \frac{1}{2}x - \frac{3}{2} + 1 \ y & = \frac{1}{2}x - \frac{3}{2} + \frac{2}{2} \y & = \frac{1}{2}x - \frac{1}{2}\end{aligned}\).

- Write the final equation in slope-intercept form

The simplified form of the equation is \( y = \frac{1}{2}x - \frac{1}{2}\), which is in the slope-intercept form \ (y = mx + b) \ where \(m = \frac{1}{2}\) and the y-intercept \(b = -\frac{1}{2}\).

Key concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to represent the equation of a line. It is written as ewline y = mx + b ewline where:

• m represents the slope of the line
• b represents the y-intercept

This form is particularly useful because it immediately tells us the slope and the y-intercept of the line. Slope, denoted by m, indicates how steep the line is. The y-intercept, denoted by b, is where the line crosses the y-axis.

point-slope formula

The point-slope formula is another way of writing the equation of a line. It is given by: ewline y - y₁ = m(x - x₁) ewline where:

• (x₁, y₁) is a known point on the line
• m is the slope of the line

This formula is especially handy when you have a point and the slope, and you need to find the equation of the line. In the example given, substituting the point (3,1) and slope ⅟1/2 yields: ewline y - 1 = ⅟1/2(x - 3).

slope of a line

The slope of a line is a measure of how steep a line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. More formally, it is given by: ewline m = (y₂ - y₁) / (x₂ - x₁) The slope can be positive, negative, zero, or undefined:

• A positive slope means the line rises as it moves from left to right
• A negative slope means the line falls as it moves from left to right
• A zero slope means the line is horizontal
• An undefined slope means the line is vertical

In this exercise, the slope is ⅟1/2, indicating a gentle, positive incline.

equation of a line

The equation of a line describes all the points (x,newline y) that lie on the line. Depending on the information available, you can write it in various forms:

• Slope-intercept form: y = mx + b
• Point-slope form: y - y₁ = m(x - x₁)
• Standard form: Ax + By = C, where A, B, and C are integers

In this case, given the slope ⅟1/2 and the point (3,1), we use the point-slope form to find the slope-intercept form of the line: y - 1 = ⅟1/2(x - 3) ewline After simplifying, we get the slope-intercept form: y = ⅟1/2x - ⅟1/2.