Solve The System By Graphing:$ \begin{align*} y &= \frac{1}{2}x + 4 \ y &= -\frac{5}{2}x - 2 \end{align*} }$Select The Correct Response A. { (-2, 3)$ $
Introduction
Solving systems of equations is a fundamental concept in algebra and mathematics. There are various methods to solve systems of equations, including substitution, elimination, and graphing. In this article, we will focus on solving systems of equations by graphing. This method involves graphing the two equations on the same coordinate plane and finding the point of intersection, which represents the solution to the system.
What is a System of Equations?
A system of equations is a set of two or more equations that contain the same variables. In this case, we have two linear equations with two variables, x and y. The first equation is y = (1/2)x + 4, and the second equation is y = (-5/2)x - 2.
Graphing the Equations
To graph the equations, we need to find the x-intercepts and y-intercepts of each equation. The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.
For the first equation, y = (1/2)x + 4, we can find the x-intercept by setting y = 0 and solving for x.
0 = (1/2)x + 4
Subtracting 4 from both sides gives:
-4 = (1/2)x
Multiplying both sides by 2 gives:
-8 = x
So, the x-intercept of the first equation is (-8, 0).
To find the y-intercept, we can set x = 0 and solve for y.
y = (1/2)(0) + 4
Simplifying gives:
y = 4
So, the y-intercept of the first equation is (0, 4).
For the second equation, y = (-5/2)x - 2, we can find the x-intercept by setting y = 0 and solving for x.
0 = (-5/2)x - 2
Adding 2 to both sides gives:
2 = (-5/2)x
Multiplying both sides by 2 gives:
4 = -5x
Dividing both sides by -5 gives:
-4/5 = x
So, the x-intercept of the second equation is (-4/5, 0).
To find the y-intercept, we can set x = 0 and solve for y.
y = (-5/2)(0) - 2
Simplifying gives:
y = -2
So, the y-intercept of the second equation is (0, -2).
Graphing the Equations on the Coordinate Plane
Now that we have found the x-intercepts and y-intercepts of each equation, we can graph the equations on the coordinate plane.
The first equation, y = (1/2)x + 4, has an x-intercept of (-8, 0) and a y-intercept of (0, 4). The graph of this equation is a line that passes through these two points.
The second equation, y = (-5/2)x - 2, has an x-intercept of (-4/5, 0) and a y-intercept of (0, -2). The graph of this equation is a line that passes through these two points.
Finding the Point of Intersection
To find the point of intersection, we need to find the point where the two lines intersect. This can be done by finding the x-coordinate of the point of intersection and then substituting this value into one of the equations to find the y-coordinate.
Let's find the x-coordinate of the point of intersection. We can do this by setting the two equations equal to each other and solving for x.
(1/2)x + 4 = (-5/2)x - 2
Adding 5/2x to both sides gives:
(1/2)x + 5/2x + 4 = -2
Combining like terms gives:
3/2x + 4 = -2
Subtracting 4 from both sides gives:
3/2x = -6
Multiplying both sides by 2/3 gives:
x = -4
So, the x-coordinate of the point of intersection is -4.
Now, we can substitute this value into one of the equations to find the y-coordinate. Let's use the first equation.
y = (1/2)x + 4
Substituting x = -4 gives:
y = (1/2)(-4) + 4
Simplifying gives:
y = 2
So, the point of intersection is (-4, 2).
Conclusion
Solving systems of equations by graphing is a useful method for finding the solution to a system of linear equations. By graphing the two equations on the same coordinate plane and finding the point of intersection, we can determine the solution to the system. In this article, we have seen how to graph the equations, find the point of intersection, and solve the system of equations.
Final Answer
The correct answer is:
- (-4, 2)
Solving Systems of Equations by Graphing: Q&A =====================================================
Introduction
In our previous article, we discussed how to solve systems of equations by graphing. This method involves graphing the two equations on the same coordinate plane and finding the point of intersection, which represents the solution to the system. In this article, we will answer some frequently asked questions about solving systems of equations by graphing.
Q: What are the advantages of solving systems of equations by graphing?
A: The advantages of solving systems of equations by graphing include:
- It is a visual method that allows you to see the relationship between the two equations.
- It is a simple method that requires minimal calculations.
- It is a good method for solving systems of linear equations.
Q: What are the disadvantages of solving systems of equations by graphing?
A: The disadvantages of solving systems of equations by graphing include:
- It may not be accurate if the graphs are not drawn correctly.
- It may not be suitable for solving systems of non-linear equations.
- It may not be suitable for solving systems of equations with multiple solutions.
Q: How do I graph the equations on the coordinate plane?
A: To graph the equations on the coordinate plane, follow these steps:
- Find the x-intercepts and y-intercepts of each equation.
- Plot the x-intercepts and y-intercepts on the coordinate plane.
- Draw a line through the x-intercepts and y-intercepts to form the graph of each equation.
- Find the point of intersection by finding the x-coordinate of the point of intersection and then substituting this value into one of the equations to find the y-coordinate.
Q: How do I find the point of intersection?
A: To find the point of intersection, follow these steps:
- Set the two equations equal to each other and solve for x.
- Substitute the value of x into one of the equations to find the value of y.
- The point of intersection is the point (x, y) that satisfies both equations.
Q: What are some common mistakes to avoid when solving systems of equations by graphing?
A: Some common mistakes to avoid when solving systems of equations by graphing include:
- Not drawing the graphs accurately.
- Not finding the point of intersection correctly.
- Not checking for multiple solutions.
- Not using a ruler or other straightedge to draw the graphs.
Q: Can I use graphing to solve systems of non-linear equations?
A: No, graphing is not suitable for solving systems of non-linear equations. Non-linear equations have graphs that are not straight lines, and graphing may not be able to accurately represent the relationship between the two equations.
Q: Can I use graphing to solve systems of equations with multiple solutions?
A: No, graphing is not suitable for solving systems of equations with multiple solutions. Graphing will only find one solution, and you may need to use other methods to find the other solutions.
Conclusion
Solving systems of equations by graphing is a useful method for finding the solution to a system of linear equations. By graphing the two equations on the same coordinate plane and finding the point of intersection, we can determine the solution to the system. In this article, we have answered some frequently asked questions about solving systems of equations by graphing.
Final Answer
The correct answer is:
- (-4, 2)