Question

Which of the following numbers is the slope of the line \(3 x-2 y+12=0 ?\) \(\begin{array}{llll}{\text { i. } 6} & {\text { ii. } 3} & {\text { iii. } 3 / 2} & {\text { iv. } 2 / 3}\end{array}\)

Short answer

The slope of the line \(3x - 2y + 12 = 0\) is \(\frac{3}{2}\). So, the correct answer is option iii.

Step-by-step solution

Rewrite the equation

First, the given equation should be rewritten into the slope-intercept form, i.e., y = mx + b. Start with \(3x - 2y + 12 = 0\). Isolate 'y' by subtracting '3x' and '-12' from both sides, which gives \(-2y = -3x - 12\).

Solve for y

Next, divide every term by '-2' to isolate 'y'. The resulting equation is \(y = 1.5x + 6\).

Identify the slope

Now that the equation is in slope-intercept form, it is apparent that the number multiplied by 'x' is the slope of the line. Therefore, the slope of this equation is 1.5 or \(\frac{3}{2}\).

Key concepts

Slope-Intercept Form

When it comes to understanding straight lines on a graph, the slope-intercept form is incredibly useful. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is the point where the line crosses the y-axis.

Using the slope-intercept form allows us to quickly identify key characteristics of a line. The slope tells us how steep the line is and in which direction it tilts—upward or downward. Positive slopes tilt upwards, while negative slopes tilt downwards. In the case of the given exercise, the equation was transformed into slope-intercept form, revealing the slope directly as part of the equation.

Linear Equations

Linear equations are fundamental in algebra and they represent straight lines on a graph. A standard linear equation might not always start in the form \( y = mx + b \), but it can be manipulated to take this shape. The equation \( 3x - 2y + 12 = 0 \) from the exercise is a prime example of this.

Linear equations reflect a consistent rate of change—meaning, for each unit increase in \( x \), \( y \) increases by a fixed amount, the slope. In this exercise, we had to apply algebraic operations to rearrange the equation into the slope-intercept form to find that consistent rate of change, which turned out to be \( \frac{3}{2} \).

Isolating Variables

Isolating variables is a critical skill in algebra, which allows us to solve for a specific variable. It involves performing operations such as addition, subtraction, multiplication, and division to get the variable by itself on one side of the equation.

Step by step, we conduct these operations inversely to the order of operations—working from outside in. In the given problem, we subtracted and then divided to isolate \( y \). It's like unraveling a knot: each action helps to free the variable, step by step, until it's clearly solved as in \( y = 1.5x + 6 \), which is neat and tidy for analysis or graphing.