2. Use Technology To Graph The System Of Equations. State Any Solutions. Round Each Coordinate To The Nearest Tenth, If Necessary.$\[ \begin{cases} y = \frac{4}{5}x - 5 \\ y = X^2 - 4x - 5 \end{cases} \\]A. (0, -5)B. (0, -5) And (4.8, -1.2)C.

Introduction

Graphing systems of equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the intersection points of two or more equations, which can be linear or non-linear. In this article, we will focus on graphing a system of two equations, one linear and one quadratic, and state any solutions.

The System of Equations

The given system of equations is:

{ \begin{cases} y = \frac{4}{5}x - 5 \\ y = x^2 - 4x - 5 \end{cases} \}

Graphing the Linear Equation

The first equation is a linear equation in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $\frac{4}{5}$ and the y-intercept is $-5$. To graph this equation, we can use the slope-intercept form and plot two points on the coordinate plane.

Graphing the Quadratic Equation

The second equation is a quadratic equation in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the equation is $y = x^2 - 4x - 5$. To graph this equation, we can use the vertex form and plot two points on the coordinate plane.

Graphing the System of Equations

To graph the system of equations, we need to find the intersection points of the two equations. We can do this by setting the two equations equal to each other and solving for $x$. Once we have the value of $x$, we can substitute it into one of the original equations to find the corresponding value of $y$.

Solving for x

To solve for $x$, we can set the two equations equal to each other:

$\frac{4}{5}x - 5 = x^2 - 4x - 5$

We can simplify this equation by multiplying both sides by $5$:

$4x - 25 = 5x^2 - 20x - 25$

We can then rearrange the equation to get a quadratic equation in the form of $ax^2 + bx + c = 0$:

$5x^2 - 24x = 0$

We can factor out an $x$ from the equation:

$x(5x - 24) = 0$

This gives us two possible values for $x$: $x = 0$ and $x = \frac{24}{5}$.

Solving for y

Now that we have the value of $x$, we can substitute it into one of the original equations to find the corresponding value of $y$. Let's use the first equation:

$y = \frac{4}{5}x - 5$

Substituting $x = 0$, we get:

$y = \frac{4}{5}(0) - 5$

$y = -5$

Substituting $x = \frac{24}{5}$, we get:

$y = \frac{4}{5}(\frac{24}{5}) - 5$

$y = \frac{96}{25} - 5$

$y = \frac{96 - 125}{25}$

$y = -\frac{29}{25}$

However, we can simplify this further by converting the fraction to a decimal and rounding to the nearest tenth:

$y = -1.2$

State the Solutions

The solutions to the system of equations are:

• $(0, -5)$
• $(4.8, -1.2)$

Conclusion

$$$$

Introduction

Graphing systems of equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we explored how to graph a system of two equations, one linear and one quadratic, and state any solutions. In this article, we will answer some frequently asked questions about graphing systems of equations.

Q: What is the purpose of graphing systems of equations?

A: The purpose of graphing systems of equations is to find the intersection points of two or more equations, which can be linear or non-linear. This can help us solve algebraic problems and understand the relationships between different variables.

Q: How do I graph a system of equations?

A: To graph a system of equations, you can use the following steps:

1. Graph each equation separately on the coordinate plane.
2. Find the intersection points of the two equations by setting them equal to each other and solving for x.
3. Substitute the value of x into one of the original equations to find the corresponding value of y.

Q: What are some common types of systems of equations?

A: Some common types of systems of equations include:

• Linear systems: These are systems of linear equations, where each equation is in the form of y = mx + b.
• Quadratic systems: These are systems of quadratic equations, where each equation is in the form of y = ax^2 + bx + c.
• Non-linear systems: These are systems of non-linear equations, where each equation is not in the form of y = mx + b or y = ax^2 + bx + c.

Q: How do I use technology to graph a system of equations?

A: There are several ways to use technology to graph a system of equations, including:

• Graphing calculators: These are handheld devices that can graph equations and find intersection points.
• Computer algebra systems: These are software programs that can graph equations and find intersection points.
• Online graphing tools: These are web-based tools that can graph equations and find intersection points.

Q: What are some common mistakes to avoid when graphing systems of equations?

A: Some common mistakes to avoid when graphing systems of equations include:

• Not setting the equations equal to each other to find the intersection points.
• Not substituting the value of x into one of the original equations to find the corresponding value of y.
• Not using technology to graph the system of equations, which can lead to errors and inaccuracies.

Q: How do I check my work when graphing a system of equations?

A: To check your work when graphing a system of equations, you can use the following steps:

1. Graph the system of equations using technology.
2. Find the intersection points of the two equations.
3. Substitute the value of x into one of the original equations to find the corresponding value of y.
4. Check that the values of x and y are consistent with the graph.

Conclusion

Graphing systems of equations is a powerful tool for solving algebraic problems. By understanding the purpose of graphing systems of equations, how to graph them, and common mistakes to avoid, you can become more confident and proficient in graphing systems of equations. Remember to use technology to graph the system of equations and check your work to ensure accuracy and precision.