Solve The Following System Of Equations Graphically On The Set Of Axes Below.${ \begin{array}{c} y=-\frac{1}{3}x+3 \ 3x-y=7 \end{array} }$Plot Two Lines By Clicking The Graph. Click A Line To Delete It.

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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. One of the methods to solve a system of equations is by graphing the equations on a coordinate plane. In this article, we will discuss how to solve a system of equations graphically using the given equations.

The Given Equations

The given system of equations is:

{ \begin{array}{c} y=-\frac{1}{3}x+3 \\ 3x-y=7 \end{array} \}

The first equation is in slope-intercept form, where the slope is −13-\frac{1}{3} and the y-intercept is 3. The second equation can be rewritten in slope-intercept form as y=3x−7y=3x-7.

Graphing the Equations

To graph the equations, we need to find the x and y intercepts of each equation.

Graphing the First Equation

The x-intercept of the first equation is found by setting y=0y=0 and solving for xx.

0=−13x+30=-\frac{1}{3}x+3

−13x=−3-\frac{1}{3}x=-3

x=9x=9

The y-intercept of the first equation is given as 3.

Graphing the Second Equation

The x-intercept of the second equation is found by setting y=0y=0 and solving for xx.

0=3x−70=3x-7

3x=73x=7

x=73x=\frac{7}{3}

The y-intercept of the second equation is found by setting x=0x=0 and solving for yy.

y=3(0)−7y=3(0)-7

y=−7y=-7

Plotting the Lines

To plot the lines, we need to find two points on each line. We can use the x and y intercepts to find these points.

For the first equation, we have the points (9, 0) and (0, 3).

For the second equation, we have the points (73,0)\left(\frac{7}{3}, 0\right) and (0, -7).

Solving the System of Equations

To solve the system of equations, we need to find the point of intersection of the two lines. We can do this by setting the two equations equal to each other and solving for xx.

−13x+3=3x−7-\frac{1}{3}x+3=3x-7

−13x−3x=−7−3-\frac{1}{3}x-3x=-7-3

−103x=−10-\frac{10}{3}x=-10

x=3x=3

Now that we have the value of xx, we can substitute it into one of the original equations to find the value of yy.

y=−13(3)+3y=-\frac{1}{3}(3)+3

y=−1+3y=-1+3

y=2y=2

Therefore, the solution to the system of equations is (3, 2).

Conclusion

In this article, we discussed how to solve a system of equations graphically using the given equations. We graphed the equations, found the x and y intercepts, and plotted the lines. We then solved the system of equations by finding the point of intersection of the two lines. The solution to the system of equations is (3, 2).

Discussion

  • What are some other methods to solve a system of equations?
  • How can we use technology to solve a system of equations?
  • What are some real-world applications of solving a system of equations?

References

  • [1] "Solving Systems of Equations" by Math Open Reference
  • [2] "Graphing Systems of Equations" by Purplemath
  • [3] "Solving Systems of Equations" by Khan Academy

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Introduction

In our previous article, we discussed how to solve a system of equations graphically using the given equations. In this article, we will answer some frequently asked questions about solving a system of equations graphically.

Q&A

Q: What are some other methods to solve a system of equations?

A: There are several other methods to solve a system of equations, including:

  • Substitution Method: This method involves substituting one equation into the other equation to solve for the variables.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
  • Graphing Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
  • Matrix Method: This method involves using matrices to solve the system of equations.

Q: How can we use technology to solve a system of equations?

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

  • Graphing Calculators: These calculators can be used to graph the equations and find the point of intersection.
  • Computer Algebra Systems: These systems can be used to solve the system of equations using various methods, including substitution, elimination, and matrix methods.
  • Online Graphing Tools: These tools can be used to graph the equations and find the point of intersection.

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

A: Solving a system of equations has many real-world applications, including:

  • Physics: Solving a system of equations is used to describe the motion of objects in physics.
  • Engineering: Solving a system of equations is used to design and optimize systems in engineering.
  • Economics: Solving a system of equations is used to model economic systems and make predictions about the economy.
  • Computer Science: Solving a system of equations is used in computer science to solve problems in computer graphics, game development, and more.

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

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

  • Not plotting the lines correctly: Make sure to plot the lines correctly and find the point of intersection.
  • Not using a ruler or straightedge: Use a ruler or straightedge to draw the lines and find the point of intersection.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions and eliminate any solutions that are not valid.

Q: How can we check if our solution is correct?

A: To check if our solution is correct, we can:

  • Plug the solution into both equations: Make sure that the solution satisfies both equations.
  • Check if the solution is a point of intersection: Make sure that the solution is a point of intersection of the two lines.
  • Use a calculator or computer algebra system: Use a calculator or computer algebra system to check if the solution is correct.

Conclusion

In this article, we answered some frequently asked questions about solving a system of equations graphically. We discussed various methods to solve a system of equations, including substitution, elimination, and matrix methods. We also discussed how to use technology to solve a system of equations and some real-world applications of solving a system of equations.

Discussion

  • What are some other methods to solve a system of equations?
  • How can we use technology to solve a system of equations?
  • What are some real-world applications of solving a system of equations?

References

  • [1] "Solving Systems of Equations" by Math Open Reference
  • [2] "Graphing Systems of Equations" by Purplemath
  • [3] "Solving Systems of Equations" by Khan Academy