Using The Slope And The \[$ Y \$\]-intercept, Graph The Line Represented By The Following Equation:$\[ 2x - Y + 4 = 0 \\]Complete Your Work In The Space Provided Or Upload A File That Can Display Math Symbols If Your Work Requires It.

Understanding the Equation

The given equation is in the form of a linear equation, which is $2x - y + 4 = 0$. To graph this line, we need to find the slope and the y-intercept of the line. The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Converting the Equation to Slope-Intercept Form

To convert the given equation to slope-intercept form, we need to isolate the variable $y$ on one side of the equation. We can do this by subtracting $2x$ from both sides of the equation and then adding $4$ to both sides.

$$2x - y + 4 = 0$$

$$-y = -2x - 4$$

$$y = 2x + 4$$

Now that we have the equation in slope-intercept form, we can identify the slope and the y-intercept.

Identifying the Slope and Y-Intercept

The slope of the line is the coefficient of the $x$ term, which is $2$. The y-intercept is the constant term, which is $4$.

Graphing the Line

To graph the line, we can use the slope and the y-intercept to find two points on the line. We can then draw a line through these two points to represent the line.

Finding Two Points on the Line

We can find two points on the line by substituting different values of $x$ into the equation and solving for $y$. Let's find the points where $x = 0$ and $x = 1$.

Point 1: $x = 0$

Substituting $x = 0$ into the equation, we get:

$$y = 2(0) + 4$$

$$y = 4$$

So, the point where $x = 0$ is $(0, 4)$.

Point 2: $x = 1$

Substituting $x = 1$ into the equation, we get:

$$y = 2(1) + 4$$

$$y = 6$$

So, the point where $x = 1$ is $(1, 6)$.

Drawing the Line

Now that we have two points on the line, we can draw a line through these two points to represent the line. The line will have a slope of $2$ and a y-intercept of $4$.

Conclusion

In this article, we learned how to graph a line using the slope and the y-intercept. We converted the given equation to slope-intercept form, identified the slope and the y-intercept, and found two points on the line. We then drew a line through these two points to represent the line. This method can be used to graph any line in slope-intercept form.

Key Takeaways

• The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• To graph a line, we need to find the slope and the y-intercept of the line.
• We can find the slope and the y-intercept by converting the equation to slope-intercept form.
• We can find two points on the line by substituting different values of $x$ into the equation and solving for $y$.
• We can draw a line through these two points to represent the line.

Common Mistakes

• Not converting the equation to slope-intercept form before finding the slope and the y-intercept.
• Not finding two points on the line before drawing the line.
• Not using a ruler or a straightedge to draw the line.

Real-World Applications

• Graphing a line is an important skill in mathematics and science.
• It is used to represent the relationship between two variables.
• It is used to make predictions and model real-world situations.

Practice Problems

• Graph the line represented by the equation $x + 2y - 3 = 0$.
• Find the slope and the y-intercept of the line represented by the equation $y = 3x - 2$.
• Graph the line represented by the equation $y = 2x + 1$.

Conclusion

Frequently Asked Questions

Q: What is the slope-intercept form of a linear equation? A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I convert a linear equation to slope-intercept form? A: To convert a linear equation to slope-intercept form, you need to isolate the variable $y$ on one side of the equation. You can do this by subtracting the $x$ term from both sides of the equation and then adding the constant term to both sides.

Q: What is the slope of a line? A: The slope of a line is the coefficient of the $x$ term in the slope-intercept form of the equation. It represents the rate of change of the line.

Q: What is the y-intercept of a line? A: The y-intercept of a line is the constant term in the slope-intercept form of the equation. It represents the point where the line intersects the y-axis.

Q: How do I find two points on a line? A: To find two points on a line, you can substitute different values of $x$ into the equation and solve for $y$. You can then use these points to draw the line.

Q: How do I draw a line through two points? A: To draw a line through two points, you can use a ruler or a straightedge to draw a line that passes through both points.

Q: What are some common mistakes to avoid when graphing a line? A: Some common mistakes to avoid when graphing a line include not converting the equation to slope-intercept form before finding the slope and the y-intercept, not finding two points on the line before drawing the line, and not using a ruler or a straightedge to draw the line.

Q: What are some real-world applications of graphing a line? A: Some real-world applications of graphing a line include representing the relationship between two variables, making predictions, and modeling real-world situations.

Q: How do I practice graphing a line? A: You can practice graphing a line by working through practice problems, such as graphing the line represented by the equation $x + 2y - 3 = 0$ or finding the slope and the y-intercept of the line represented by the equation $y = 3x - 2$.

Q: What are some tips for graphing a line? A: Some tips for graphing a line include using a ruler or a straightedge to draw the line, labeling the x and y axes, and using a grid to help you draw the line.

Q: Can I graph a line using a calculator? A: Yes, you can graph a line using a calculator. Many calculators have a graphing function that allows you to enter the equation and view the graph.

Q: How do I know if my graph is correct? A: To check if your graph is correct, you can use a ruler or a straightedge to draw a line through the two points you found and compare it to the graph you drew. If the two lines are the same, then your graph is correct.

Q: What are some common errors to look out for when graphing a line? A: Some common errors to look out for when graphing a line include drawing the line through the wrong points, not using a ruler or a straightedge to draw the line, and not labeling the x and y axes.

Conclusion

In conclusion, graphing a line using the slope and the y-intercept is an important skill in mathematics and science. By following the steps outlined in this article and practicing graphing a line, you can become proficient in graphing lines and apply this skill to real-world situations.