Problem 53 - Thinking Mathematically

Question

In Exercises 53-54, find 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 \(-A/B\) and the y-intercept is \(C/B\).

Step 1: Find the slope

The 'slope' (m) of a line in the equation form \(Ax + By = C\) is given by \(-A/B\). So, substitute the values of A and B into this formula to get the slope.

Step 2: Find the y-intercept

The 'y-intercept' (b) of a line in the equation form \(Ax + By = C\) is given by \(C/B\). Substitute the values of C and B into this formula to get the y-intercept.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness or incline. In linear equations, understanding how to calculate the slope is essential for comprehending how different variables interact on a graph. The slope is often denoted by the letter 'm'. When working with equations in the standard form, represented by \(Ax + By = C\), the formula to find the slope is

• \(m = -\frac{A}{B}\)

In this expression, 'A' and 'B' are coefficients from the equation. Remember to pay attention to the negative sign; it's crucial because it affects the direction of the slope.
The slope tells us how much the line rises or falls as you move along the x-axis. For instance:

• A positive slope means that the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope means the line is horizontal.

Understanding how to correctly apply the slope formula will help you analyze and graph linear equations with ease.

Y-Intercept Determination

The y-intercept is the point where the line crosses the y-axis on a graph. In terms of an equation, it's the value of 'y' when 'x' is zero. Finding the y-intercept is straightforward once you understand how it fits into the linear equation in standard form.

• For \(Ax + By = C\), the y-intercept 'b' is given by \(b = \frac{C}{B}\).

Here, 'C' and 'B' are constants from the equation. Simply divide 'C' by 'B' to find where the line will intersect the y-axis.
The y-intercept isn't just a mere number; it's an essential aspect that helps us draw the whole graph accurately. For any line equation, knowing the slope and y-intercept allows you to sketch the line entirely without needing more points.

Linear Equations

Linear equations form the foundation of algebraic expressions and are vital in high school mathematics and beyond. These equations represent straight lines when graphed on a coordinate plane. Linear equations can be written in different forms, but the standard form is:

• \(Ax + By = C\)

Where 'A', 'B', and 'C' are constants and both 'A' and 'B' cannot be zero simultaneously.
Linear equations have multiple characteristics:

• The graph of a linear equation is always a straight line.
• They have a constant slope, meaning the inclination of the line never changes.
• They help solve numerous real-world problems by modeling relationships between quantities.

Understanding linear equations involves knowing how to convert between different forms and interpret their graphical representations. Remember, every linear equation helps to define how two variables relate in a proportionate manner.