Question

A system of linear equations is shown below. $$ \begin{aligned} 3 y &=-2 x+8 \\ 3 y &=2 x+8 \end{aligned} $$ Which of the following describes the graph of this system of linear equations in the standard \((x, y)\) coordinate plane? A. Two distinct intersecting lines B. Two parallel lines with positive slope C. Two parallel lines with negative slope D. A single line with positive slope E. A single line with negative slope

Short answer

A. Two distinct intersecting lines B. Two parallel lines with positive slope C. Two parallel lines with negative slope D. A single line with positive slope E. A single line with negative slope

Step-by-step solution

Identify the given linear equations

The given system of linear equations is: $$ \begin{aligned} 3 y &= -2 x + 8 \\ 3 y &= 2 x + 8 \end{aligned} $$

Rewrite the equations in slope-intercept form

To understand the nature of the graphs of these equations and their slopes, rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. For the first equation, divide both sides by 3: $$y = -\frac{2}{3}x + \frac{8}{3}$$ For the second equation, divide both sides by 3: $$y = \frac{2}{3}x + \frac{8}{3}$$

Compare the slopes and y-intercepts

Now we can compare the slopes (m) and y-intercepts (b) of the two equations. For the first equation: Slope, \(m_1 = -\frac{2}{3}\) y-intercept, \(b_1 = \frac{8}{3}\) For the second equation: Slope, \(m_2 = \frac{2}{3}\) y-intercept, \(b_2 = \frac{8}{3}\) Since the slopes are not equal (\(m_1 \neq m_2\)), the lines are not parallel. The y-intercepts are equal (\(b_1 = b_2\)), but this does not affect whether the lines intersect, because the lines have different slopes.

Identify the characteristics of the system of linear equations

As mentioned before, the lines are not parallel. They have different slopes and will intersect at some point. Both lines have a negative and positive slope: First equation: Negative slope Second equation: Positive slope

Choose the correct answer

Now we can compare our findings with the given options and choose the correct one: A. Two distinct intersecting lines B. Two parallel lines with positive slope C. Two parallel lines with negative slope D. A single line with positive slope E. A single line with negative slope The correct answer is A. Two distinct intersecting lines.

Key concepts

Slope-Intercept Form

The slope-intercept form of a line is critical for understanding linear equations and how they are represented on a graph. It is expressed as

\( y = mx + b \)

where \( m \) stands for the slope of the line, which shows the steepness and the direction, while \( b \) represents the y-intercept—the point where the line crosses the y-axis. This form is particularly convenient because it gives you the visual layout of the line immediately; a positive slope tilts upwards, while a negative slope tilts downwards.

Application in Exercise

To utilize this form in practice, like in the given exercise, the equations first need to be manipulated into the slope-intercept form to easily compare their slopes and y-intercepts. These characteristics determine the behavior of the graphs and their potential intersection points. In the example, rewriting the given equations into the slope-intercept form revealed distinct slopes but identical y-intercepts, leading to the conclusion that our graphs show two intersecting lines.

Graphing Linear Equations

Visualizing linear equations by graphing them on the coordinate plane is a foundational skill in algebra. Graphing involves plotting points and drawing lines based on the equation's slope and y-intercept. The graph provides a picture of all possible (x, y) solutions to the equation.

The y-intercept is where you start on the graph; it's where the line crosses the y-axis. The slope tells you how to move from this point: a slope of \( m \) means that for each step right (increase in x), you move \( m \) steps up or down (depending on whether the slope is positive or negative).

Exercise Visualization

In our exercise, graphing both equations after converting them to slope-intercept form would reveal their point of intersection and confirm that the lines cross, showcasing two distinct lines. This visual assessment supports the algebraic conclusion that we have intersecting lines, matching answer A in the exercise.

Comparing Slopes

Slopes are pivotal in determining the nature of the relationship between two lines when comparing linear equations. Two key rules about slopes help identify whether lines are parallel, intersecting, or the same line:

• If two lines have the same slope, they are parallel.
• If they have different slopes, they will intersect at some point.

A zero slope means the line is horizontal, while an undefined slope (division by zero) indicates a vertical line.

Insights from Our Equations

Upon comparison in our exercise, one line had a positive slope and the other had a negative slope. This implies not only will the lines intersect, but they will do so in such a way that forms a clear 'X' shape on the graph. They are mirror images in terms of slope, but because they share the same y-intercept, they intersect exactly at that y-intercept, thus confirming that the correct answer is A, which signifies that the lines are distinct and do intersect.