Solve This System Of Equations By Graphing. First, Graph The Equations, And Then Type The Solution.$\[ \begin{align*} y &= \frac{1}{3}x + 6 \\ y &= \frac{1}{6}x + 5 \end{align*} \\] Click To Select Points On The Graph.
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 used to solve a system of equations is graphing. Graphing involves plotting the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system. In this article, we will learn how to solve a system of equations by graphing using the given equations: and .
Graphing the Equations
To graph the equations, we need to find two points on each line. We can do this by substituting different values of x into the equation and solving for y. Let's start with the first equation: . We can substitute x = 0, 3, 6, and 9 into the equation to find the corresponding values of y.
| x | y |
|---|---|
| 0 | 6 |
| 3 | 7 |
| 6 | 8 |
| 9 | 9 |
Now, let's plot these points on a coordinate plane.
+---------------------------------------+
| |
| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| +---------------------------------------+
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| | | <br/>
**Solve this System of Equations by Graphing**
=====================================================
**Q&A: Solving Systems of Equations by Graphing**
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**Q: What is a system of equations?**
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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 graph the equations?**
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A: Graphing the equations helps us visualize the relationship between the variables and find the point of intersection, which represents the solution to the system.
**Q: How do we graph the equations?**
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A: To graph the equations, we need to find two points on each line. We can do this by substituting different values of x into the equation and solving for y.
**Q: What are the steps to graph the equations?**
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A: The steps to graph the equations are:
1. Find two points on each line by substituting different values of x into the equation and solving for y.
2. Plot the points on a coordinate plane.
3. Draw a line through the points to represent the equation.
4. Find the point of intersection between the two lines.
**Q: How do we find the point of intersection?**
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A: The point of intersection is the point where the two lines meet. To find the point of intersection, we need to set the two equations equal to each other and solve for x.
**Q: What is the solution to the system of equations?**
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A: The solution to the system of equations is the point of intersection between the two lines. This point represents the values of x and y that satisfy both equations.
**Q: How do we check the solution?**
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A: To check the solution, we need to substitute the values of x and y into both equations and make sure they are true.
**Q: What are some common mistakes to avoid when graphing the equations?**
-------------------------------------------------------------------
A: Some common mistakes to avoid when graphing the equations are:
* Not finding enough points on each line
* Not plotting the points correctly on the coordinate plane
* Not drawing the lines through the points correctly
* Not finding the point of intersection correctly
**Q: What are some tips for graphing the equations?**
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A: Some tips for graphing the equations are:
* Use a ruler to draw the lines through the points
* Use a pencil to plot the points on the coordinate plane
* Make sure to label the axes and the points correctly
* Use a calculator to check the solution
**Conclusion**
----------
Solving systems of equations by graphing is a useful technique for finding the values of the variables. By following the steps outlined above, we can graph the equations and find the point of intersection, which represents the solution to the system. Remember to check the solution by substituting the values of x and y into both equations and make sure they are true.
**Example**
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Let's use the example given in the introduction to solve the system of equations by graphing.
```markdown
y = 1/3x + 6
y = 1/6x + 5
To graph the equations, we need to find two points on each line. We can do this by substituting different values of x into the equation and solving for y.
| x | y |
|---|---|
| 0 | 6 |
| 3 | 7 |
| 6 | 8 |
| 9 | 9 |
Now, let's plot these points on a coordinate plane.
+---------------------------------------+
| |
| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| +---------------------------------------+
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