Find The Solution To The Following System Of Equations By Graphing:${ \begin{cases} y = 6 - \frac{2}{3}x \ y = X + 1 \end{cases} }$

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of equations by graphing, which is a visual method that uses graphs to find the solution.

What is a System of Equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is an equation that contains one or more variables, and the system is solved by finding the values of the variables that satisfy all the equations in the system.

Graphing a System of Equations

Graphing a system of equations involves graphing each equation in the system on a coordinate plane and finding the point of intersection of the two graphs. The point of intersection is the solution to the system of equations.

Step 1: Graph the First Equation

The first equation in the system is $y = 6 - \frac{2}{3}x$. To graph this equation, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

The slope of the first equation is $-\frac{2}{3}$, which is a negative slope. This means that the graph of the first equation will be a line that slopes downward from left to right.

The y-intercept of the first equation is 6, which means that the graph of the first equation will pass through the point (0, 6).

To graph the first equation, we can use the slope-intercept form of a linear equation and plot the point (0, 6). We can then use the slope to find other points on the graph.

Step 2: Graph the Second Equation

The second equation in the system is $y = x + 1$. To graph this equation, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

The slope of the second equation is 1, which is a positive slope. This means that the graph of the second equation will be a line that slopes upward from left to right.

The y-intercept of the second equation is 1, which means that the graph of the second equation will pass through the point (0, 1).

To graph the second equation, we can use the slope-intercept form of a linear equation and plot the point (0, 1). We can then use the slope to find other points on the graph.

Step 3: Find the Point of Intersection

To find the point of intersection of the two graphs, we need to find the point where the two lines intersect.

To find the point of intersection, we can set the two equations equal to each other and solve for x.

$6 - \frac{2}{3}x = x + 1$

We can then solve for x by isolating x on one side of the equation.

$6 - 1 = x + \frac{2}{3}x$

$5 = \frac{5}{3}x$

$x = 3$

Now that we have found the value of x, we can substitute this value into one of the original equations to find the value of y.

$y = 6 - \frac{2}{3}(3)$

$y = 6 - 2$

$y = 4$

Therefore, the point of intersection of the two graphs is (3, 4).

Conclusion

In this article, we have solved a system of equations by graphing. We have graphed each equation in the system on a coordinate plane and found the point of intersection of the two graphs. The point of intersection is the solution to the system of equations.

Tips and Tricks

• When graphing a system of equations, it is helpful to use a coordinate plane with a grid to make it easier to plot the points.
• When finding the point of intersection, it is helpful to use the substitution method to solve for x.
• When solving a system of equations, it is helpful to check the solution by plugging it back into each of the original equations.

Real-World Applications

Solving a system of equations by graphing has many real-world applications. For example, in physics, a system of equations can be used to model the motion of an object. In economics, a system of equations can be used to model the behavior of a market. In engineering, a system of equations can be used to design a system.

Common Mistakes

• When graphing a system of equations, it is easy to make mistakes by plotting the points incorrectly.
• When finding the point of intersection, it is easy to make mistakes by solving for x incorrectly.
• When solving a system of equations, it is easy to make mistakes by not checking the solution.

Conclusion

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: Why do we need to solve a system of equations?

A: We need to solve a system of equations to find the values of the variables that satisfy all the equations in the system. This is useful in many real-world applications, such as physics, economics, and engineering.

Q: What is the graphing method for solving a system of equations?

A: The graphing method for solving a system of equations involves graphing each equation in the system on a coordinate plane and finding the point of intersection of the two graphs.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the point of intersection?

A: The point of intersection is the point where the two graphs intersect. This is the solution to the system of equations.

Q: How do I find the point of intersection?

A: To find the point of intersection, you can set the two equations equal to each other and solve for x. You can then substitute this value into one of the original equations to find the value of y.

Q: What are some common mistakes to avoid when solving a system of equations by graphing?

A: Some common mistakes to avoid when solving a system of equations by graphing include:

• Plotting the points incorrectly
• Solving for x incorrectly
• Not checking the solution

Q: What are some real-world applications of solving a system of equations by graphing?

A: Some real-world applications of solving a system of equations by graphing include:

• Modeling the motion of an object in physics
• Modeling the behavior of a market in economics
• Designing a system in engineering

Q: Can I use the graphing method to solve a system of equations with more than two equations?

A: Yes, you can use the graphing method to solve a system of equations with more than two equations. However, it may be more difficult to find the point of intersection of multiple graphs.

Q: Can I use a calculator to graph a system of equations?

A: Yes, you can use a calculator to graph a system of equations. Many graphing calculators have a built-in function to graph a system of equations.

Q: What are some tips for graphing a system of equations?

A: Some tips for graphing a system of equations include:

• Using a coordinate plane with a grid to make it easier to plot the points
• Using the slope-intercept form of a linear equation to graph the equations
• Checking the solution by plugging it back into each of the original equations

Q: Can I use the graphing method to solve a system of equations with non-linear equations?

A: No, the graphing method is typically used to solve systems of linear equations. Non-linear equations may require other methods, such as substitution or elimination, to solve.

Q: What are some common types of systems of equations that can be solved by graphing?

A: Some common types of systems of equations that can be solved by graphing include:

• Systems of linear equations
• Systems of quadratic equations
• Systems of polynomial equations

Q: Can I use the graphing method to solve a system of equations with complex numbers?

A: Yes, you can use the graphing method to solve a system of equations with complex numbers. However, it may be more difficult to graph complex numbers on a coordinate plane.