Graph The Line. Y + 5 = − 1 2 ( X + 1 Y + 5 = -\frac{1}{2}(x + 1 Y + 5 = − 2 1 ( X + 1 ]

Mar 9, 2025 by ADMIN 91 views

**Graphing the Line: A Step-by-Step Guide**

Understanding the Equation

The given equation is in the form of a linear equation, which is a fundamental concept in mathematics. The equation is $y + 5 = -\frac{1}{2}(x + 1)$ . To graph this line, we need to understand the components of the equation and how they relate to the graph.

Slope-Intercept Form

The equation can be rewritten in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. To do this, we need to isolate $y$ on one side of the equation.

Isolating y

To isolate $y$ , we need to get rid of the fraction and the constant term on the right-hand side of the equation. We can do this by multiplying both sides of the equation by 2, which is the denominator of the fraction.

2(y + 5) = -\frac{1}{2}(x + 1) \times 2

This simplifies to:

2y + 10 = -x - 1

Simplifying the Equation

Now, we can simplify the equation by getting rid of the constant term on the right-hand side. We can do this by adding 1 to both sides of the equation.

2y + 10 + 1 = -x - 1 + 1

This simplifies to:

2y + 11 = -x

Isolating y

Now, we can isolate $y$ by subtracting 11 from both sides of the equation and then dividing both sides by 2.

2y = -x - 11

y = \frac{-x - 11}{2}

Slope-Intercept Form

Now that we have isolated $y$ , we can rewrite the equation in slope-intercept form.

y = -\frac{1}{2}x - \frac{11}{2}

Graphing the Line

To graph the line, we need to find the x-intercept and the y-intercept. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

Finding the x-Intercept

To find the x-intercept, we need to set $y$ equal to 0 and solve for $x$ .

0 = -\frac{1}{2}x - \frac{11}{2}

\frac{11}{2} = -\frac{1}{2}x

x = -11

Finding the y-Intercept

To find the y-intercept, we need to set $x$ equal to 0 and solve for $y$ .

y = -\frac{1}{2}(0) - \frac{11}{2}

y = -\frac{11}{2}

Graphing the Line

Now that we have found the x-intercept and the y-intercept, we can graph the line. The x-intercept is (-11, 0) and the y-intercept is (0, -11/2).

Graphing the Line: A Conclusion

In conclusion, graphing the line $y + 5 = -\frac{1}{2}(x + 1)$ requires us to understand the components of the equation and how they relate to the graph. We can rewrite the equation in slope-intercept form, isolate $y$ , and then find the x-intercept and the y-intercept. By graphing the line, we can visualize the relationship between the variables and gain a deeper understanding of the equation.

Key Takeaways

The equation $y + 5 = -\frac{1}{2}(x + 1)$ can be rewritten in slope-intercept form as $y = -\frac{1}{2}x - \frac{11}{2}$ .
To graph the line, we need to find the x-intercept and the y-intercept.
The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.
By graphing the line, we can visualize the relationship between the variables and gain a deeper understanding of the equation.

Real-World Applications

Graphing lines is a fundamental concept in mathematics that has numerous real-world applications. Some examples include:

Physics: Graphing lines is used to model the motion of objects and predict their trajectories.
Engineering: Graphing lines is used to design and optimize systems, such as bridges and buildings.
Economics: Graphing lines is used to model economic systems and predict trends in the market.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about graphing the line $y + 5 = -\frac{1}{2}(x + 1)$ .

Q: What is the slope of the line?

A: The slope of the line is -1/2. This can be determined by looking at the coefficient of x in the equation.

Q: What is the y-intercept of the line?

A: The y-intercept of the line is -11/2. This can be determined by looking at the constant term in the equation.

Q: How do I find the x-intercept of the line?

A: To find the x-intercept of the line, set y equal to 0 and solve for x. This will give you the point where the line crosses the x-axis.

Q: How do I find the y-intercept of the line?

A: To find the y-intercept of the line, set x equal to 0 and solve for y. This will give you the point where the line crosses the y-axis.

Q: Can I graph the line by hand?

A: Yes, you can graph the line by hand. To do this, plot the x-intercept and the y-intercept on a coordinate plane and then draw a line through these two points.

Q: Can I use a graphing calculator to graph the line?

A: Yes, you can use a graphing calculator to graph the line. To do this, enter the equation into the calculator and then use the graphing function to visualize the line.

Q: What is the equation of the line in slope-intercept form?

A: The equation of the line in slope-intercept form is y = -1/2x - 11/2.

Q: Can I use the equation of the line to solve real-world problems?

A: Yes, you can use the equation of the line to solve real-world problems. For example, you can use the equation to model the motion of an object or to predict the trajectory of a projectile.

Common Mistakes to Avoid

When graphing the line $y + 5 = -\frac{1}{2}(x + 1)$ , there are several common mistakes to avoid.

Mistake 1: Not rewriting the equation in slope-intercept form

A: Failing to rewrite the equation in slope-intercept form can make it difficult to graph the line.

Mistake 2: Not finding the x-intercept and the y-intercept

A: Failing to find the x-intercept and the y-intercept can make it difficult to graph the line.

Mistake 3: Not using a graphing calculator or graphing by hand

A: Failing to use a graphing calculator or graphing by hand can make it difficult to visualize the line.

Conclusion

In conclusion, graphing the line $y + 5 = -\frac{1}{2}(x + 1)$ requires a clear understanding of the equation and how it relates to the graph. By rewriting the equation in slope-intercept form, finding the x-intercept and the y-intercept, and using a graphing calculator or graphing by hand, you can visualize the line and gain a deeper understanding of the equation.