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. Include The

$$

Understanding the Equation

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

Rewriting the Equation

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

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

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

$$y = 2x + 4$$

Identifying the Slope and $y$-Intercept

Now that we have the equation in slope-intercept form, we can identify the slope and $y$-intercept. The slope is the coefficient of $x$, 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 $y$-intercept to find two points on the line. We can then draw a line through these two points to graph the line.

Finding the $y$-Intercept

The $y$-intercept is the point where the line intersects the $y$-axis. Since the $y$-intercept is $4$, we can plot the point $(0, 4)$ on the coordinate plane.

Finding a Second Point

To find a second point on the line, we can use the slope to find the $x$-coordinate of the point. Since the slope is $2$, we can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Let's choose the point $(0, 4)$ as the point $(x_1, y_1)$. We can then plug in the values of $m$, $x_1$, and $y_1$ into the point-slope form:

$$y - 4 = 2(x - 0)$$

$$y - 4 = 2x$$

$$y = 2x + 4$$

Now that we have the equation in point-slope form, we can find the $x$-coordinate of the second point. Let's choose $x = 1$. We can then plug in the value of $x$ into the equation:

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

$$y = 2 + 4$$

$$y = 6$$

So, the second point is $(1, 6)$.

Graphing the Line

Now that we have two points on the line, we can graph the line by drawing a line through these two points. 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 $y$-intercept. We started by rewriting the equation in slope-intercept form, which allowed us to identify the slope and $y$-intercept. We then used the slope and $y$-intercept to find two points on the line, which we used to graph the line. The line has a slope of $2$ and a $y$-intercept of $4$.

Example Problems

Problem 1

Graph the line represented by the equation $3x - 2y + 5 = 0$.

Solution

To graph the line, we need to rewrite the equation in slope-intercept form. We can do this by subtracting $3x$ from both sides and adding $2y$ to both sides:

$$3x - 2y + 5 = 0$$

$$-2y = -3x - 5$$

$$y = \frac{3}{2}x + \frac{5}{2}$$

The slope is $\frac{3}{2}$ and the $y$-intercept is $\frac{5}{2}$. We can use these values to find two points on the line. Let's choose the point $(0, \frac{5}{2})$ as the $y$-intercept. We can then use the slope to find the $x$-coordinate of the second point. Let's choose $x = 1$. We can then plug in the value of $x$ into the equation:

$$y = \frac{3}{2}(1) + \frac{5}{2}$$

$$y = \frac{3}{2} + \frac{5}{2}$$

$$y = \frac{8}{2}$$

$$y = 4$$

So, the second point is $(1, 4)$. We can then graph the line by drawing a line through these two points.

Problem 2

Graph the line represented by the equation $x + 2y - 3 = 0$.

Solution

To graph the line, we need to rewrite the equation in slope-intercept form. We can do this by subtracting $x$ from both sides and adding $2y$ to both sides:

$$x + 2y - 3 = 0$$

$$2y = -x + 3$$

$$y = -\frac{1}{2}x + \frac{3}{2}$$

The slope is $-\frac{1}{2}$ and the $y$-intercept is $\frac{3}{2}$. We can use these values to find two points on the line. Let's choose the point $(0, \frac{3}{2})$ as the $y$-intercept. We can then use the slope to find the $x$-coordinate of the second point. Let's choose $x = 1$. We can then plug in the value of $x$ into the equation:

$$y = -\frac{1}{2}(1) + \frac{3}{2}$$

$$y = -\frac{1}{2} + \frac{3}{2}$$

$$y = \frac{2}{2}$$

$$y = 1$$

So, the second point is $(1, 1)$. We can then graph the line by drawing a line through these two points.

Final Thoughts

Graphing a line using the slope and $y$-intercept is a useful skill to have in mathematics. It allows us to visualize the line and understand its properties. In this article, we learned how to graph a line using the slope and $y$-intercept, and we saw how to apply this skill to example problems. With practice and patience, you can become proficient in graphing lines using the slope and $y$-intercept.

$$

Understanding the Basics

Graphing a line using the slope and $y$-intercept is a fundamental concept in mathematics. It allows us to visualize the line and understand its properties. In this article, we will answer some common questions about graphing a line using the slope and $y$-intercept.

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 find the slope and $y$-intercept of a linear equation?

A: To find the slope and $y$-intercept of a linear equation, you need to rewrite the equation in slope-intercept form. This can be done by subtracting $x$ from both sides and adding $y$ to both sides.

Q: What is the $y$-intercept?

A: The $y$-intercept is the point where the line intersects the $y$-axis. It is the value of $y$ when $x = 0$.

Q: How do I graph a line using the slope and $y$-intercept?

A: To graph a line using the slope and $y$-intercept, you need to find two points on the line. You can do this by using the slope to find the $x$-coordinate of the second point, and then plugging in the value of $x$ into the equation to find the $y$-coordinate.

Q: What is the significance of the slope in graphing a line?

A: The slope is the rate of change of the line. It tells us how steep the line is and in which direction it is sloping.

Q: How do I find the $x$-coordinate of a point on a line?

A: To find the $x$-coordinate of a point on a line, you need to use the slope to find the $x$-coordinate of the point. This can be done by plugging in the value of $y$ into the equation and solving for $x$.

Q: What is the difference between the slope and the $y$-intercept?

A: The slope is the rate of change of the line, while the $y$-intercept is the point where the line intersects the $y$-axis.

Q: How do I graph a line with a negative slope?

A: To graph a line with a negative slope, you need to find two points on the line. You can do this by using the slope to find the $x$-coordinate of the second point, and then plugging in the value of $x$ into the equation to find the $y$-coordinate.

Q: What is the significance of the $y$-intercept in graphing a line?

A: The $y$-intercept is the point where the line intersects the $y$-axis. It is the value of $y$ when $x = 0$.

Q: How do I graph a line with a zero slope?

A: To graph a line with a zero slope, you need to find the $y$-intercept of the line. This can be done by plugging in the value of $x$ into the equation and solving for $y$.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to find the $x$-intercepts and the $y$-intercept of the equation. You can do this by plugging in the value of $x$ into the equation and solving for $y$.

Conclusion

Graphing a line using the slope and $y$-intercept is a fundamental concept in mathematics. It allows us to visualize the line and understand its properties. In this article, we answered some common questions about graphing a line using the slope and $y$-intercept. We hope that this article has been helpful in understanding the basics of graphing a line using the slope and $y$-intercept.

Final Thoughts

Graphing a line using the slope and $y$-intercept is a useful skill to have in mathematics. It allows us to visualize the line and understand its properties. With practice and patience, you can become proficient in graphing lines using the slope and $y$-intercept.