Problem 27 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=-4 x+5 $$

Step-by-Step Solution

Answer

Answer: The slope (m) is -4 and the y-intercept (b) is 5.

Step 1: Identify the slope-intercept form of the equation

The equation is already given in the form y = mx + b, where m is the slope and b is the y-intercept. The equation is: $$ y = -4x + 5 $$

Step 2: Determine the slope (m)

From the equation, we can see that the slope is -4. Therefore, the slope (m) is: $$ m = -4 $$

Step 3: Determine the y-intercept (b)

From the equation, we can see that the y-intercept is 5. Therefore, the y-intercept (b) is: $$ b = 5 $$ In conclusion, the slope of the line is -4, and the y-intercept is 5.

Key Concepts

Understanding the Slope

In linear equations, the slope is a crucial concept. It represents how steep a line is on a graph and the direction it moves. The slope is often denoted by the letter "m" in the slope-intercept form of a linear equation, which is written as \( y = mx + b \). In this form, "m" indicates the relationship between the changes in the y-values to changes in the x-values.

To put it simply, the slope shows how much y changes for a one-unit increase in x. If the slope is positive, the line goes upwards from left to right. If it's negative, like our example with \( m = -4 \), the line goes downwards. The larger or smaller the number, the steeper the incline or decline of the line.

• Positive slope: line increases from left to right.
• Negative slope: line decreases from left to right.
• Zero slope: flat line.
• Undefined slope: vertical line.

Recognizing the slope quickly helps in graphing linear equations efficiently.

Exploring the Y-intercept

The y-intercept is another key element in understanding linear equations. This is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by "b". It's the value of y when x is zero, giving us a fixed starting point of the line on a graph.

Knowing the y-intercept helps in plotting the line since it's the first point you can mark before using the slope to find other points on the line. For the equation \( y = -4x + 5 \), the y-intercept is 5. This means when x is 0, y equals 5, so the line crosses the y-axis at the point (0, 5).

• A positive "b" moves the line up.
• A negative "b" moves the line down.

The y-intercept gives you a direct insight into how the line is positioned vertically on a graph.

Linear Equations Made Simple

Linear equations are fundamental in mathematics. They describe the relationship between two variables, usually x and y, that can be plotted as a straight line. Understanding linear equations allows us to see how changing one variable affects another in a uniform way.

The slope-intercept form is one of the most accessible forms of a linear equation, \( y = mx + b \). It straightforwardly reveals the slope and y-intercept, making graphing easier. This relationship is always consistent: for any increase in x by a certain amount, y will increase or decrease by the slope times that amount.

• Helps predict outcomes.
• Useful in real-life situations like business trends or physics.

Linear equations and their interpretations are a stepping stone to understanding more complex mathematical concepts. They are essential tools not only in algebra but also in other fields that rely on statistical data and analysis.