Solve The System Of Linear Equations By Graphing. Round The Solution To The Nearest Tenth.$\[ \begin{array}{l} y = -0.25x + 4.7 \\ y = 4.9x - 1.64 \end{array} \\]The Approximate Solution To The System Is \[$(\square, \square)\$\].
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Introduction
Solving a system of linear equations involves finding the point of intersection between two or more lines. In this article, we will explore how to solve a system of linear equations by graphing. We will use two linear equations in the form of y = mx + b, where m is the slope and b is the y-intercept. The system of linear equations is given by:
{ \begin{array}{l} y = -0.25x + 4.7 \\ y = 4.9x - 1.64 \end{array} \}
Understanding the Graphing Method
The graphing method involves graphing both lines on the same coordinate plane and finding the point of intersection. This point represents the solution to the system of linear equations. To graph the lines, 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.
Finding Points on the First Line
Let's find two points on the first line, y = -0.25x + 4.7. We can substitute x = 0 and x = 10 into the equation to find the corresponding y-values.
- For x = 0, y = -0.25(0) + 4.7 = 4.7
- For x = 10, y = -0.25(10) + 4.7 = 3.3
So, the two points on the first line are (0, 4.7) and (10, 3.3).
Finding Points on the Second Line
Now, let's find two points on the second line, y = 4.9x - 1.64. We can substitute x = 0 and x = 10 into the equation to find the corresponding y-values.
- For x = 0, y = 4.9(0) - 1.64 = -1.64
- For x = 10, y = 4.9(10) - 1.64 = 46.36
So, the two points on the second line are (0, -1.64) and (10, 46.36).
Graphing the Lines
Now that we have the points on both lines, we can graph them on the same coordinate plane. We can use a graphing calculator or software to graph the lines.
Graphing the First Line
The first line, y = -0.25x + 4.7, is a decreasing line with a slope of -0.25. We can graph the line by plotting the two points (0, 4.7) and (10, 3.3) and drawing a line through them.
Graphing the Second Line
The second line, y = 4.9x - 1.64, is an increasing line with a slope of 4.9. We can graph the line by plotting the two points (0, -1.64) and (10, 46.36) and drawing a line through them.
Finding the Point of Intersection
Now that we have graphed both lines, we can find the point of intersection by looking for the point where the two lines cross. This point represents the solution to the system of linear equations.
Finding the x-Coordinate
To find the x-coordinate of the point of intersection, we can set the two equations equal to each other and solve for x.
-0.25x + 4.7 = 4.9x - 1.64
We can add 0.25x to both sides of the equation to get:
4.7 = 5.15x - 1.64
We can add 1.64 to both sides of the equation to get:
6.34 = 5.15x
We can divide both sides of the equation by 5.15 to get:
x = 1.23
Finding the y-Coordinate
Now that we have the x-coordinate of the point of intersection, we can substitute it into one of the original equations to find the y-coordinate. We will use the first equation, y = -0.25x + 4.7.
y = -0.25(1.23) + 4.7
We can simplify the equation to get:
y = 4.5
So, the point of intersection is (1.23, 4.5).
Conclusion
In this article, we have explored how to solve a system of linear equations by graphing. We used two linear equations in the form of y = mx + b and graphed them on the same coordinate plane. We found the point of intersection by looking for the point where the two lines crossed. The point of intersection represents the solution to the system of linear equations. We rounded the solution to the nearest tenth to get (1.2, 4.5).
Final Answer
The approximate solution to the system is (1.2, 4.5).
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Introduction
In our previous article, we explored how to solve a system of linear equations by graphing. We used two linear equations in the form of y = mx + b and graphed them on the same coordinate plane. We found the point of intersection by looking for the point where the two lines crossed. In this article, we will answer some frequently asked questions about solving systems of linear equations by graphing.
Q&A
Q: What is the graphing method for solving systems of linear equations?
A: The graphing method involves graphing both lines on the same coordinate plane and finding the point of intersection. This point represents the solution to the system of linear equations.
Q: How do I find the points on each line?
A: To find the points on each line, you can substitute different values of x into the equation and solve for y. For example, if you have the equation y = -0.25x + 4.7, you can substitute x = 0 and x = 10 into the equation to find the corresponding y-values.
Q: How do I graph the lines?
A: To graph the lines, you can plot the two points on each line and draw a line through them. You can use a graphing calculator or software to graph the lines.
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. Then, you can substitute the x-value into one of the original equations to find the y-value.
Q: What if the lines are parallel?
A: If the lines are parallel, they will never intersect. In this case, the system of linear equations has no solution.
Q: What if the lines intersect at a point that is not a whole number?
A: If the lines intersect at a point that is not a whole number, you will need to round the solution to the nearest tenth.
Q: Can I use the graphing method to solve systems of linear equations with more than two variables?
A: No, the graphing method is only suitable for solving systems of linear equations with two variables.
Q: Are there any other methods for solving systems of linear equations?
A: Yes, there are several other methods for solving systems of linear equations, including substitution, elimination, and matrices.
Conclusion
In this article, we have answered some frequently asked questions about solving systems of linear equations by graphing. We have covered topics such as finding points on each line, graphing the lines, finding the point of intersection, and more. We hope that this article has been helpful in clarifying any confusion you may have had about solving systems of linear equations by graphing.
Final Answer
The graphing method is a useful tool for solving systems of linear equations, but it is not the only method. There are several other methods available, and the choice of method will depend on the specific problem and the level of difficulty.