Problem 95 In the following exercises, grap... [FREE SOLUTION]

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Chapter 3: Problem 95

In the following exercises, graph each line with the given point and slope. $$ (-1,-4) ; m=\frac{4}{3} $$

Short Answer

Expert verified

Graph using the point-slope equation $ y = \frac{4}{3}x - \frac{8}{3} $ and the points $(-1, -4)$ and $ (2, 0) $.

Step by step solution

Identify the given point and slope

The point given is $(-1, -4)$ and the slope $m$ is $\frac{4}{3}$.

Use the point-slope form of the equation of a line

The point-slope form of a line's equation is $y - y_1 = m(x - x_1)$. Plug in the given point $(x_1, y_1) = (-1, -4)$ and the slope $m = \frac{4}{3}$. This gives $y + 4 = \frac{4}{3}(x + 1)$.

Simplify the equation to slope-intercept form

To express the equation in slope-intercept form $y = mx + b$, first distribute the $ \frac{4}{3} $ on the right side: $y + 4 = \frac{4}{3}x + \frac{4}{3}$. Next, isolate $y$ by subtracting 4 from both sides to get $y = \frac{4}{3}x + \frac{4}{3} - 4$. \-4 can be written as $-\frac{12}{3}$, so the final form is $y = \frac{4}{3}x - \frac{8}{3}$.

Plot the given point on the graph

Plot the point $(-1, -4)$ on the coordinate plane.

Use the slope to find another point

The slope $ \frac{4}{3} $ means for every 3 units you move to the right (positive x-direction), you move up 4 units (positive y-direction). Starting from $(-1, -4)$, moving 3 steps right to $x = -1 + 3 = 2$ and 4 steps up to $y = -4 + 4 = 0$, we get another point $ (2, 0) $.

Draw the line

Draw a straight line through the points $(-1, -4)$ and $ (2, 0) $ to graph the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form

The point-slope form of a linear equation is useful when you have a point and a slope. The formula is

$y - y_1 = m(x - x_1)$.

This tells you how the y-value changes with the x-value starting from a particular point $(x_1, y_1)$.
In the exercise, we start with the point $(-1, -4)$ and the slope $m = \frac{4}{3}$.
This form helps us quickly write an equation for the line

Slope-Intercept Form

The slope-intercept form is another way to write the equation of a line.
It is written as $y = mx + b$.

Here, $m$ is the slope, and $b$ is the y-intercept where the line crosses the y-axis.

To convert from point-slope form to slope-intercept form:

First, distribute the slope $m$.
Then, isolate $y$ by moving the constant term to the other side.

For example, from $y + 4 = \frac{4}{3}(x + 1)$, we distribute to get $y + 4 = \frac{4}{3}x + \frac{4}{3}$.

Then we subtract 4 from both sides to find $y$, resulting in $y = \frac{4}{3}x - \frac{8}{3}$.

Plotting Points

Plotting points is a crucial step in graphing any line. To start:

Locate the x-coordinate on the x-axis.

From this x-coordinate, move vertically to find the y-coordinate.

For instance, the point $(-1, -4)$:

Locate -1 on the x-axis.

Move down to -4 on the y-axis and place a point.

This forms the basis of the graph.

Plotting the second point using the slope helps reinforce accuracy.

With a slope of $\frac{4}{3}$:

Walk 3 steps to the right (positive x-direction).

Then rise 4 steps up (positive y-direction).

This leads to the second point $(2, 0)$.

Graphing Lines

Graphing lines involves drawing a straight line that passes through all plotted points.

To graph the line:

First, plot all your points.

Then, use a ruler to draw a straight line through them.

This line represents all solutions to your linear equation.

In our exercise, drawing a line through points $(-1, -4)$ and $(2, 0)$ shows the graph for the equation $y = \frac{4}{3}x - \frac{8}{3}$.

Always ensure your line extends through the whole graph to show the relationship between x and y across a range of values.

This gives a complete visual representation.

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91影视

In the following exercises, graph each line with the given point and slope. $$ (-1,-4) ; m=\frac{4}{3} $$

Short Answer

Step by step solution

Identify the given point and slope

Use the point-slope form of the equation of a line

Simplify the equation to slope-intercept form

Plot the given point on the graph

Use the slope to find another point

Draw the line

Key Concepts

Point-Slope Form

Slope-Intercept Form

Plotting Points

Graphing Lines

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