91Ó°ÊÓ

Parallel lines are fascinating because they never meet, no matter how far you extend them. Imagine two train tracks racing endlessly alongside each other. This happens because parallel lines share the same slope, meaning they rise and run at the exact same rate.
In geometry, if you know the slope of one line, finding a parallel line requires using the same slope. The equation for the given line is already in the slope-intercept form: \( y = mx + b \). Here, \( m \) represents the slope. If the equation is \( y = 3x - 2 \), the slope \( m \) is 3.
When lines are parallel, they preserve this slope value. So, for any new line parallel to \( y = 3x - 2 \), you use the same slope, 3.
Here’s the magic touch: if you want to find another line that is parallel to a given line and passes through a particular point, you keep the slope constant, ensuring they remain perfectly parallel.

The point-slope form is a very handy tool in algebra. It's especially useful when you know a line's slope and a specific point on it. It can be thought of as a bridge to reaching different forms of a line's equation. The point-slope formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is that crucial point, and \( m \) is the slope.
Here's how it works: if you have the point \( \left( \frac{2}{3}, \frac{1}{4} \right) \) and you've already determined the slope is 3, finding the equation becomes straightforward. Plug the point values and slope into the formula:

• \( y - \frac{1}{4} = 3 \left( x - \frac{2}{3} \right) \)

This setup quickly leads you to the equation of the line. It’s a solid starting point before transitioning into the slope-intercept form.

Every line on a graph can be described with a precise equation, most commonly expressed in the slope-intercept form: \( y = mx + b \). This form gives you a clear view of the line’s slope \( m \) and where it cuts through the y-axis \( b \).
To find an equation of a line parallel to a given one, first extract the slope. For instance, from \( y = 3x - 2 \), we see the slope is 3. Then, using the point-slope setup, apply this slope:

• Begin with: \( y - \frac{1}{4} = 3 \left( x - \frac{2}{3} \right) \)
• Distribute and rearrange to reach the traditional slope-intercept form:

Solve by distributing and adding constant terms to both sides to arrive finally at the simplified slope-intercept form of the line passing through your given point:

• \( y = 3x - \frac{7}{4} \)

This method ensures you track the line's gradient and its specific crossing of the y-axis, which is vital for drawing or analyzing the line.