Problem 35 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -3 y=12 x-27 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -4, and the y-intercept is 9.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the given equation, \(-3y = 12x - 27\), in slope-intercept form, we need to solve for y. $$ -3y = 12x - 27 $$ Divide both sides by -3. $$ y = -4x + 9 $$ Now the equation is in slope-intercept form.

Step 2: Identify the slope

The slope-intercept form of a line equation is y = mx + b, where m is the slope. Comparing our equation, \(y = -4x + 9\), to the slope-intercept form, we can see that: $$ m = -4 $$ So, the slope of the given line is -4.

Step 3: Identify the y-intercept

In the slope-intercept form of a line equation, the y-intercept, b, is the value of y when x is 0. Inspecting our equation, \(y = -4x + 9\), we can see that the y-intercept is: $$ b = 9 $$ So, the y-intercept of the given line is 9.

Step 4: State the slope and y-intercept

The slope of the given line equation, \(-3y = 12x - 27\), is -4 and the y-intercept is 9.

Key Concepts

Slope Determination

The process of finding the slope of a line, called slope determination, involves understanding the basic concept of slope from the equation of a line. The slope is a measure of how steep a line is. It indicates the change in the vertical direction (rise) in relation to the change in the horizontal direction (run). When we express a linear equation in the slope-intercept form, which is \(y = mx + b\), the slope \(m\) is the coefficient of \(x\). For the equation we derived, \(y = -4x + 9\), our slope \(m\) equals -4. This means for every one unit increase in \(x\) along the horizontal axis, \(y\) decreases by 4 units on the vertical axis.

• The slope provides insight into the direction of the line: a positive slope rises, while a negative slope falls.
• A slope of zero indicates a horizontal line.
• Understanding the slope helps in predicting the behavior of a line graphically.

Y-Intercept Calculation

Calculating the y-intercept from a linear equation is straightforward once the equation is in slope-intercept form, \(y = mx + b\). The y-intercept is the value of \(y\) when \(x\) is zero. In the equation \(y = -4x + 9\), the y-intercept, \(b\), is simply the constant term, which is 9. This means the line crosses the y-axis at the point (0, 9).

• The y-intercept provides a starting point for graphing a line, making it crucial for visual representations.
• In context, it often represents the initial condition or starting value in practical problems.
• It’s crucial for formulating the whole picture of a line together with the slope.

Linear Equations

Linear equations are fundamental in algebra, describing a straight line on a graph. Each linear equation can be transformed into the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.The original given equation, \(-3y = 12x - 27\), is an example of a linear equation. By rearranging this equation into the slope-intercept form, we were able to find the slope and y-intercept, which are essential for graphing and understanding the line.

• Linear equations often come from real-world relationships, like calculating cost, speed, or predicting profit.
• Transforming them into slope-intercept form simplifies understanding and analysis.
• They serve as a basis for exploring more complex mathematical concepts and graphs.