Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (6,2), m=\frac{1}{2} $$

Short answer

The equation of the line in point-slope form: \(y - 2 = \frac{1}{2}(x - 6)\). In slope-intercept form: \(y = \frac{1}{2}x - 1\).

Step-by-step solution

Write in Point-Slope Form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where m is the slope of the line and \( (x_1,y_1) \) are the coordinates of a point on the line. Substitute the point (6, 2) and the slope \( \frac{1}{2} \) into the formula: \(y - 2 = \frac{1}{2}(x - 6)\).

Simplify

In this step, distribute \( \frac{1}{2} \) to both \(x\) and \(-6\). The simplified equation becomes \( y - 2 = \frac{1}{2}x - 3 \).

Write in Slope-Intercept Form

The slope-intercept form of the equation of a line is \(y = mx + b\), where m is the slope of the line and b is the intercept. To transform our simplified equation into this form, add 2 to both sides: \(y = \frac{1}{2}x - 1 \).

Key concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest and most commonly used ways to write the equation of a line. It is represented as:
\( y = mx + b \),
where \( m \) is the slope of the line, indicating its steepness and direction, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful when you need to quickly identify these two key characteristics of the line.

To convert the point-slope form to the slope-intercept form, you'll need to solve for y. Start with the point-slope equation:
\( y - y_1 = m(x - x_1) \),
and then manipulate it to isolate \( y \) on one side of the equation. Simplify and adjust the equation until it resembles the slope-intercept form.

Equation of a Line

The equation of a line is a formula that describes all the points that make up the line. Several forms exist, each offering different benefits depending on what information you have. Apart from the slope-intercept form \( y = mx + b \), there's also the point-slope form:
\( y - y_1 = m(x - x_1) \),
which is useful when you're given a point \( (x_1, y_1) \) on the line and the slope \( m \).

Take the example from the exercise where the point \( (6,2) \) and the slope \( \frac{1}{2} \) were given. By substituting these into the point-slope formula, we quickly get to the equation of the line. This versatility makes it easier for students to tackle various problems involving linear equations.

Linear Equations

Linear equations form the foundation for much of algebra and are equations that make straight lines when graphed. They follow the standard form \( Ax + By = C \), the slope-intercept form \( y = mx + b \), or the point-slope form \( y - y_1 = m(x - x_1) \). The fundamental characteristic of these equations is their constant slope, meaning they increase or decrease at a uniform rate.

When solving linear equations, remember to perform the same operations on both sides of the equation to maintain balance. The process standardly involves simplifying terms, isolating the variable, and sometimes converting between different forms to make use of specific information you may have about the line, such as a point or slope.