Problem 78 - Precalculus Essentials

Question

Give the slope and y-intercept of each line whose equation is given. Assume that \(B \neq 0\). $$A x=B y-C$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{A}{B}\) and the y-intercept is \(\frac{C}{B}\).

Step 1: Rewrite the given equation

Let's rearrange the given equation \(Ax = By - C\) to the standard format \(y = mx + b\). We can do this by first dividing all terms by \(B\) to isolate \(y\). The rearranged equation is \[y = \frac{A}{B} x + \frac{C}{B}\].

Step 2: Identify the slope

The coefficient of \(x\) in the equation \(y = mx + b\) is the slope. In our formula, the coefficient of \(x\) is \(\frac{A}{B}\), so this is the slope, \(m\).

Step 3: Identify the y-intercept

The constant in the equation \(y = mx + b\) is the y-intercept. In our formula, the constant is \(\frac{C}{B}\), so this is the y-intercept, \(b\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when it comes to analyzing linear equations. Think of the slope as the steepness of a hill; it shows how much the line goes up or down as it moves from left to right. In mathematical language, the slope is represented by the variable 'm' and can be determined by the ratio of the change in y (vertical change) to the change in x (horizontal change), which is also known as 'rise over run'.

In the given exercise, the equation was transformed into the slope-intercept form, which is written as \(y = mx + b\). The coefficient of \(x\), which is \(\frac{A}{B}\) after rearranging the given equation, represents the slope of the line. So if you see an equation where \(x\) is multiplied by a fraction or a number, that fraction or number is your slope!

Y-Intercept

Moving on to another fundamental concept, the y-intercept is the point where the line crosses the y-axis. You can view this as the starting point of the line if you were to follow it from left to right. In the slope-intercept form of a linear equation, \(y = mx + b\), the y-intercept is denoted by 'b'.

In our example, after we have rearranged the equation to the slope-intercept form, \(\frac{C}{B}\) is what you would see at the very end, and this represents our y-intercept. It's the point \((0, \frac{C}{B})\) on the y-axis. This point is essential for graphing the line and understanding its position on the coordinate plane.

Linear Equation Standard Form

What is Standard Form?

The standard form of a linear equation is often given as \(Ax + By = C\), where A, B, and C are integers, and A should be non-negative. It's another way to neatly package the information of a line, showing its slope and y-intercept indirectly.

Converting to Slope-Intercept Form

To find the slope and y-intercept, we usually convert the standard form to slope-intercept form by solving for y. This was exemplified in the step-by-step solution of the exercise where the given equation \(Ax = By - C\) was rearranged to show y on its own on one side of the equation. This conversion is a useful technique to make the equation more intuitive for solving and graphing purposes.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. For a single linear equation, the solution is a point on the line it represents. When we have the slope-intercept form, we are well-equipped to solve the equation and graph it.

To solve a linear equation in slope-intercept form, you'll often substitute a value for x and then solve for y to find a specific point. Or, if given a value for y, you'll solve for x. The beauty of the slope-intercept form is that it tells you the starting point (the y-intercept, 'b') and the tilt of the line (the slope, 'm'), which makes graphing or finding other solutions much more straightforward.