Problem 24

Question

Write an equation for the line with Slope -5 and $y$ -intercept -2

Step-by-Step Solution

Verified

Answer

The equation of the line with slope -5 and $y$-intercept -2 is $y = -5x -2$

1Step 1: Identify the given slope and $y$-intercept

We are given that the slope ($m))$ = -5 and the $y$-intercept ($b$) = -2.

2Step 2: Substitute the slope and $y$-intercept into the slope-intercept equation

Substitute -5 for $m$ and -2 for $b$ in the equation $y = mx + b$. This gives us $y = -5x -2$

Key Concepts

Slope-Intercept FormSlopeY-intercept

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the easiest and most widely used ways to express a line equation. It allows you to quickly identify the slope and the y-intercept, which play crucial roles in defining the line's direction and position. The standard format for this form is:

$y = mx + b$

Here, $m$ represents the slope of the line, and $b$ is the y-intercept. In practical terms, the slope-intercept form allows us to see how steep a line is and where it crosses the y-axis. Understanding the slope-intercept form is essential for graphing a line quickly, making it an indispensable tool in algebra.

Slope

The slope is a fundamental concept in understanding the behavior of a line. It tells us how steep the line is and whether it rises or falls as it moves from left to right. Mathematically, the slope is represented by $m$ in the slope-intercept form and can be calculated using two points on a line:

$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$

A positive slope indicates the line rises as it moves to the right, whereas a negative slope means it falls. A steeper line has a higher absolute value of the slope. In our equation, the slope $-5$ tells us that for every step to the right, the line drops down 5 units, showing a steep downward direction.

Y-intercept

The y-intercept is the point where a line crosses the y-axis of a graph. This is when the value of $x$ is zero. In the slope-intercept form $y = mx + b$, the y-intercept is expressed by $b$. It provides essential information about where the line starts on the y-axis. Understanding the y-intercept is particularly useful when you are setting up or verifying a line graph, as it instantly tells you the line's starting point on the vertical axis. For the equation $y = -5x - 2$, the y-intercept is $-2$, meaning the line crosses the y-axis at -2. This gives a clear picture of the line's vertical starting point and helps in sketching the graph.

Problem 24

Problem 25

Other exercises in this chapter

Problem 24

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Problem 24

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Problem 25

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Problem 25

Find an equation for the line that passes through the point (2,-3) and is parallel to the $y$ -axis.

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