Question

graph the two equations in the same coordinate plane. Use the following information to decide whether the lines are perpendicular. It can be shown that two different nonvertical lines in the same plane with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{2}\) is the negative reciprocal of \(m_{1}\) $$ \begin{aligned} &y=-\frac{1}{3} x-4\\\ &y=3 x+2 \end{aligned} $$

Short answer

Yes, the lines are perpendicular as the slope of the second line is the negative reciprocal of the slope of the first.

Step-by-step solution

- Identify the slope of each line

First, look at the two given equations and identify the slope on each of them. The first equation is \(y=−\frac{1}{3}x−4\) , means the slope \(m_{1}\) is –1⁄3. The second equation is \(y=3x+2\), thus the slope \(m_{2}\) is 3.

- Check if the slopes are negatives reciprocals of each other

The negative reciprocal of a number can be obtained by first taking the reciprocal of the number, and then changing its sign. The reciprocal of –1⁄3 is –3. Changing the sign gives 3, which matches the slope \(m_{2}\) of the second line, fulfilling the condition for the lines to be perpendicular.

- Conclusion

Since slope \(m_{2}\) equals the negative reciprocal of slope \(m_{1}\), as stated in the instructions, we conclude that the two lines are perpendicular.

Key concepts

Slope of a Line

The slope of a line in mathematics represents the steepness and the direction of the line on the coordinate plane. It’s a measure of how much the line rises (or falls) as you move along it from left to right. The slope is usually represented by the letter 'm' and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line, often expressed as 'rise over run'.

The equation of a straight line is often presented in the format of y = mx + b, where 'm' stands for the slope and 'b' represents the y-intercept, the point where the line crosses the y-axis. For instance, in the equation y = −1/3x − 4, the slope 'm' is −1/3, indicating that for every unit you move to the right along the x-axis, the line goes down by 1/3 of a unit.

Negative Reciprocal

Understanding the concept of a negative reciprocal is crucial in determining the relationship between two lines, especially when looking for perpendicularity. The negative reciprocal of a given non-zero number is simply the inverse of that number, with the opposite sign. For example, if you start with the slope 2, its reciprocal is 1/2 since 2 multiplied by 1/2 equals 1. To get the negative reciprocal, you would then change the sign, making it -1/2.

Mathematically, if you have a slope m_1, the negative reciprocal is -1/m_1. In context with our exercise, the slope −1/3 has a negative reciprocal of 3 because: −1/(−1/3) = 3. This is pivotal for understanding perpendicular lines because, by definition, lines are perpendicular if one’s slope is the negative reciprocal of the other's.

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves using a pair of perpendicular lines called axes. The horizontal line is known as the x-axis, and the vertical line is the y-axis. They intersect at a point called the origin, which has coordinates (0,0). Every point on the plane is identified by an ordered pair of numbers (x, y), representing its horizontal and vertical distances from the origin.

The significance of the coordinate plane in graphing cannot be overstated—it is essential for visually representing algebraic equations. By plotting points that satisfy the equation of a line and connecting them, one can graph the line on the coordinate plane. This is a powerful way to understand the relationship between two variables represented by the x and y-axes.

Graphing Linear Equations

Graphing linear equations involves drawing a straight line that represents all solutions to the equation. The process typically includes finding two or more points that satisfy the equation and then connecting these points with a straight line. The y-intercept, where the line crosses the y-axis, is a common starting point because it's easily found from the equation format y = mx + b. Another helpful point is the x-intercept, where the line crosses the x-axis, by setting y to zero and solving for x.

Using the equations from our exercise, y = −1/3x − 4 and y = 3x + 2, you would plot the y-intercepts (-4 and 2, respectively) and use the slope to find other points. For the first equation, from the intercept, you would move down 1 unit and to the right 3 units to find your next point. Alternatively, for the second equation, from the y-intercept 2, you would move up 3 units and to the right 1 unit for another point since the slope is 3. Joining these points for each equation with a straight line would give you the graphs of the two lines.