Problem 23

Question

Determine the slope $m$ and $y$ intercept $b$ of the line with the given equation. Then sketch the line. $y-1=2(x+3)$

Step-by-Step Solution

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Answer

Slope $m$ is 2 and $y$-intercept $b$ is 7. Sketch the line starting at (0, 7) with a slope of 2.

1Step 1: Rewrite the equation in slope-intercept form

First, we convert the given equation $y - 1 = 2(x + 3)$ into the slope-intercept form, which is $y = mx + b$. To achieve this, start by distributing the 2 on the right-hand side: $y - 1 = 2x + 6$. Then, solve for $y$ by adding 1 to both sides: $y = 2x + 7$.

2Step 2: Identify the slope $m$ and $y$-intercept $b$

Now that the equation is in slope-intercept form $y = mx + b$, identify the slope $m$ and $y$-intercept $b$. Here, $m = 2$ and $b = 7$.

3Step 3: Sketch the graph of the line

To sketch the line $y = 2x + 7$, start at the $y$-intercept $b = 7$ on the $y$-axis. From there, use the slope $m = 2$, which means you rise 2 units and run 1 unit to the right for the next point on the line. Plot this second point and draw a straight line through these points, extending it in both directions.

Key Concepts

SlopeY-InterceptLinear Equations

Slope

In the context of linear equations, the slope is a key characteristic that indicates the steepness and direction of a line on a graph. The slope is denoted by the letter $ m $. It is calculated as the "rise" over the "run," which means the change in the vertical direction (y-axis) divided by the change in the horizontal direction (x-axis). In equations stated in the slope-intercept form, $ y = mx + b $, the slope is directly represented by $ m $. For example, in the equation $ y = 2x + 7 $, the slope $ m $ is $ 2 $.

Positive slope: The line rises from left to right.
Negative slope: The line falls from left to right.
Zero slope: The line is horizontal.
Undefined slope: The line is vertical, which isn't typically represented in slope-intercept form.

Understanding the slope helps in predicting how the line will look and behave on the graph. For each unit increase in $ x $, the value of $ y $ changes by the slope value. In our example, for every 1 unit increase in $ x $, $ y $ increases by 2 units.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In slope-intercept form $ y = mx + b $, the y-intercept is represented by $ b $. This is the value of $ y $ when $ x $ is zero. It is an important concept because it provides a starting point for graphing a line.

The y-intercept helps in locating where the line will cross the y-axis.
It plays a crucial role in drafting the initial part of the line on the graph.
The y-intercept is independent of the slope.

For instance, in the equation $ y = 2x + 7 $, the y-intercept $ b $ is $ 7 $. This means that the line will cross the y-axis at the point $ (0, 7) $. Knowing this allows you to start graphing the line at this point and then use the slope to find other points.

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines. They are often written in the form $ y = mx + b $, known as the slope-intercept form. This form is advantageous because it immediately provides the slope and the y-intercept, making graphing straightforward.

Linear equations form the foundation of algebra and coordinate geometry.
They can define real-world relationships with constant rates of change.
Slope-intercept form is especially useful for visualizing how variables relate to each other.

To rewrite other forms of linear equations (such as point-slope or standard form) into slope-intercept form, simple algebraic manipulations are needed. This makes various analysis tasks easier, including graphing lines, comparing slopes, or solving systems of equations. An example of converting a standard equation to slope-intercept form starts with an equation like $ Ax + By = C $, which you can rearrange to $ y = mx + b $. This friendly format helps to quickly identify the slope and y-intercept of the line.

Problem 23

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