Problem 53 - Algebra 1: Concepts and Skills

Question

Determine whether the graphs of the two equations are parallel lines. Explain your answer. $$line\quad a: y+6 x-8=0 line\quad b: 2 y=12 x-4$$

Step-by-Step Solution

Answer

No, because the slopes of the lines are not equal. Therefore, the lines are not parallel.

Step 1: Convert line a to slope-intercept form

Line a is given by the equation \(y + 6x - 8 = 0\). First, re-arrange it to y = mx + c form: -6x + y = 8, which gives y = -6x + 8.

Step 2: Convert line b to slope-intercept form

Line b is given by the equation \(2y = 12x - 4\). Again, re-arrange this to y = mx + c form by dividing through by 2: y = 6x - 2.

Step 3: Compare the slopes of the two lines

From Step 1, the slope of line a is -6 and from Step 2, the slope of line b is 6. Since the slopes are not equal, the lines are not parallel.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never intersect. They are always the same distance apart, making them look like train tracks or stripes on a road. When we're dealing with linear equations, how do we know if two lines are parallel? It's all about the slope!

Each straight line on a graph has a slope, which describes how steep it is. If two lines have the same slope, they will never meet, no matter how far they are extended. This is the mathematical condition for parallel lines.

• Slope is consistent for both lines.
• They never intersect or cross.
• Visual appearance: equal steepness.

So, in mathematics, we simply check the slopes of two lines to see if they're parallel. If the slopes are equal, the lines are parallel.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most useful forms. It's written as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. The y-intercept is where the line crosses the y-axis.

Why is this form so helpful? Because it tells us everything we need to know about the line's direction and place!

• Slope \((m)\): It shows how steep a line is. Positive means it goes up, negative means it goes down.
• Y-intercept \((c)\): The point where the line hits the y-axis. It tells us the starting point of the line.

When you rearrange an equation into this format, both the slope and intercept become immediately clear, making it easier to graph or compare it to other equations.

Graphing Equations

Graphing equations is about plotting lines on a coordinate plane. This visual representation helps us see the relationship between variables. It's like a roadmap for equations, showing us how x and y change together.

To graph a linear equation like \( y = mx + c \), follow these steps:

• Start by finding the y-intercept \(c\). This is where you'll plot the first point on the y-axis.
• Use the slope \(m\) to determine the direction of the line. For example, a slope of 2 means you go up 2 units and right 1 unit from the y-intercept.
• Draw the line by connecting the points. Extend it in both directions.

This approach is straightforward and works for any linear equation. It shows how equations translate into graphs, giving us a clearer picture of the relationships involved.