Problem 49 - Algebra 1: Concepts and Skills

Question

Determine whether the graphs of the two equations are parallel lines. Explain your answer. $$line\quad a: y=-3 x+2\quad line\quad b: y+3 x=-4$$

Step-by-Step Solution

Answer

The two lines are parallel because they have the same slope, which is -3.

Step 1: Rewriting Equation of line a

The first equation is already in slope-intercept form \(y = mx + b\). So, its slope \(m1\) is equal -3.

Step 2: Rewriting Equation of line b

The second equation is not in slope intercept form. We can rewrite it as \(y = mx + b\) by subtracting \(3x\) from both sides. This leaves us with the equation of line b as \(y = -3x - 4\). Therefore its slope, \(m2\), is -3.

Step 3: Comparing the slopes

If the slopes of two lines are equal, then the lines are parallel. Our calculations show that \(m1 = m2 = -3\). Therefore, the two lines are parallel.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when working with linear equations. This form is written as:\[ y = mx + b \] where:

• \(y\) is the dependent variable (typically the vertical axis on graphs).
• \(m\) represents the slope of the line, which indicates its steepness or slant.
• \(x\) is the independent variable (usually the horizontal axis).
• \(b\) signifies the y-intercept, where the line crosses the y-axis.

The advantage of using the slope-intercept form is its simplicity, allowing you to quickly read off both the slope and the y-intercept by simply looking at the equation. It is especially helpful in determining the nature of the graph of the equation without needing to plot it.

Equations

An equation in mathematics represents a statement where two expressions are set equal to each other. It often involves variables and constants. Understanding how to manipulate and interpret equations is key in algebra. Consider the two equations given:

• Equation of line a: \( y = -3x + 2 \) -- This is already in slope-intercept form.
• Equation of line b: \( y + 3x = -4 \) -- Needs to be rearranged to find its slope (\( m \)).

For line b, rearrange by subtracting \(3x\) from both sides to put it into slope-intercept form as:\[ y = -3x - 4 \] This manipulation reveals the slope \(m\) as -3. Equations like these help us understand how lines interact on a graph.

Slope Comparison

Comparing slopes is a vital step to determine the relationship between two lines. Whether they are parallel, perpendicular, or neither depends on their slopes.

• Parallel lines will have equal slopes.
• Perpendicular lines will have slopes that are negative reciprocals of each other.

In our exercise, lines have slopes \(m_1 = -3\) and \(m_2 = -3\). To determine if they are parallel, compare these slopes:

• Since \(m_1 = m_2\), the lines are parallel.

This shows how, with slope comparison, you can quickly assess the geometric relationship between different line equations.