Problem 3 Graph the equations. $$ y=\f... [FREE SOLUTION]

Chapter 7: Problem 3

Graph the equations. $$ y=\frac{2}{3} x+1 $$

Short Answer

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Question: Graph the equation $y = \frac{2}{3}x + 1$. Answer: To graph the equation, we first identify the slope and y-intercept, which are $\frac{2}{3}$ and $1$, respectively. We then plot the y-intercept point (0, 1) and use the slope to find another point (3, 3) by moving 3 units to the right and 2 units up from the y-intercept. We draw a line connecting these points, which represents the graph of the equation $y = \frac{2}{3}x + 1$. Optionally, we can check the graph's accuracy by plotting additional points.

Step by step solution

Identify the slope and y-intercept

The equation is already in slope-intercept form $y = mx + b$, where m represents the slope and b represents the y-intercept. Here, the slope $m = \frac{2}{3}$ and the y-intercept $b = 1$.

Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. In our equation, the y-intercept is $1$. So the point on the graph is $(0, 1)$.

Use the slope to find another point

The slope is $\frac{2}{3}$, which means that for every 3 units we move horizontally (either left or right), the line will go up or down by 2 units. Since we already have the y-intercept point $(0,1)$, we can move 3 units to the right and 2 units up to find another point on the graph: $(0+3, 1+2) = (3, 3)$. So, we have another point $(3, 3)$ on the graph.

Draw the line connecting the points

Now that we have two points $(0, 1)$ and $(3, 3)$, we can draw a line connecting them. This line will represent the graph of the equation $y = \frac{2}{3}x + 1$.

Check your graph with additional points (Optional)

To ensure the accuracy of your graph, you can choose more x-values, calculate the corresponding y-values, and plot them on the graph. If these points also lie on the line drawn in Step 4, your graph is correct.

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Key Concepts

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

Slope-Intercept Form

When it comes to graphing linear equations, the slope-intercept form is incredibly useful. It provides a straightforward way to express any straight line equation. The formula used is $y = mx + b$, where $m$ stands for the "slope" and $b$ stands for the "y-intercept." This form makes it easy for us to immediately identify how the line behaves and where it will cross the y-axis.

To keep it simple:

$y$ is the output or the vertical value (on the y-axis).
$x$ is the input or the horizontal value (on the x-axis).
$m$ tells you how steep the line is.
$b$ tells you where the line crosses the y-axis.

Understanding the slope-intercept form is critical because it allows you to easily pick out the slope and y-intercept, simplifying the process of graphing a line.

Plotting Points

Plotting points on a graph is a fundamental skill in mathematics that lets you visually represent equations. When you know the coordinates of a point, such as $0, 1$ or $3, 3$, you can locate it on the grid by identifying the appropriate x and y values.

Here's how to do it step-by-step:

Start at the origin, which is $0, 0$ where the x-axis and y-axis intersect.
Move horizontally to match the x-value of your point.
Then move vertically to match the y-value.
Mark the spot where these two positions meet. That's your plotted point!

For example, in the equation $y = \frac{2}{3}x + 1$, we determine that the y-intercept is $0, 1$. We plot this point by staying at the origin $0$ and moving up to 1 on the y-axis.

Slope of a Line

The slope of a line indicates how slanted the line is and tells us how much y increases when x increases by one. It is often represented as $m$ in the slope-intercept equation $y = mx + b$. Slope can be calculated by taking the change in y and dividing it by the change in x, which is the rise over the run.

Think of it like this:

"Rise" is how much you move up or down.
"Run" is how much you move left or right.

By using the slope of $\frac{2}{3}$ in our example, you know that for every 3 steps you take to the right, you go up 2 steps. This helps us find new points quickly by combining the slope with the starting point, like moving from $0, 1$ to $3, 3$ in the problem above.

Y-Intercept

The y-intercept is a key component in understanding the graph of a linear equation. It tells you the exact point where the line will cross the y-axis, which occurs when the value of $x$ is zero. In the slope-intercept form $y = mx + b$, the y-intercept is represented by $b$.

To find this on a graph, simply locate the point where the line crosses the y-axis.

This is the point $0, b$.
For example, in $y = \frac{2}{3}x + 1$, the y-intercept is $b = 1$, so the line passes through the point $0, 1$.

Knowing the y-intercept is important because it serves as a starting point for plotting your line on the graph, making the graphing process much simpler!

91影视

Graph the equations. $$ y=\frac{2}{3} x+1 $$

Short Answer

Step by step solution

Identify the slope and y-intercept

Plot the y-intercept

Use the slope to find another point

Draw the line connecting the points

Check your graph with additional points (Optional)

Key Concepts

Slope-Intercept Form

Plotting Points

Slope of a Line

Y-Intercept

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