Find the solution to the system of equations by grafting
As an AI language model, I cannot provide a graph, but I can show you how to solve the system of equations algebraically.
Here are the equations:
3x - y = 7
2x + y = 4
To solve the system using graphing, you would need to plot the two lines on a coordinate plane and find the point where they intersect. However, I will solve the system algebraically.
First, add the two equations to eliminate y:
3x - y + 2x + y = 7 + 4
5x = 11
Then, divide both sides by 5:
x = 11/5
Now that we have solved for x, we can substitute this value back into one of the original equations to find y. Let's choose the first equation:
3x - y = 7
3(11/5) - y = 7
33/5 - y = 7
Subtract 33/5 from both sides:
-y = -8/5
Finally, divide both sides by -1:
y = 8/5
Therefore, the solution to the system of equations is (11/5, 8/5).
To solve a system of equations by graphing, follow these steps:
Step 1: Write down the given system of equations.
For example, let's consider the following system of equations:
Equation 1: 2x + y = 4
Equation 2: 3x - 2y = 1
Step 2: Rearrange each equation to solve for y in terms of x (if possible).
Rearranging Equation 1: y = 4 - 2x
Rearranging Equation 2: y = (3x - 1)/2
Step 3: Plot the graph of each equation on the same coordinate plane.
To plot each equation, assign different values to x and calculate the corresponding y-values using the rearranged equations.
Equation 1:
Let x = 0
Then, y = 4 - 2(0) = 4. So, we have the point (0, 4).
Let x = 1
Then, y = 4 - 2(1) = 2. So, we have the point (1, 2).
Now plot the points (0, 4) and (1, 2) and draw a line passing through them.
Equation 2:
Let x = 0
Then, y = (3(0) - 1)/2 = -1/2. So, we have the point (0, -1/2).
Let x = 1
Then, y = (3(1) - 1)/2 = 1. So, we have the point (1, 1).
Now plot the points (0, -1/2) and (1, 1) and draw a line passing through them.
Step 4: Determine the point of intersection.
The solution to the system of equations is the point where the two lines intersect on the graph. In this case, it seems difficult to determine the exact point of intersection by graphing alone. Therefore, we can continue to the next step.
Step 5: Determine the coordinates of the intersection point.
To determine the coordinates of the intersection point more accurately, we can solve the system of equations algebraically using methods like substitution or elimination.
Using the given equations:
Equation 1: 2x + y = 4
Equation 2: 3x - 2y = 1
Let's solve it using substitution:
From Equation 1, we can write y = 4 - 2x.
Substituting this into Equation 2:
3x - 2(4 - 2x) = 1
Simplifying the equation:
3x - 8 + 4x = 1
7x - 8 = 1
7x = 1 + 8
7x = 9
x = 9/7
Now substitute x = 9/7 into Equation 1 to find y:
2(9/7) + y = 4
18/7 + y = 4
y = 4 - 18/7
y = (28 - 18)/7
y = 10/7
Hence, the solution to the system of equations is x = 9/7 and y = 10/7, or approximately x = 1.29 and y = 1.43.
Note: The step-by-step bot could only provide general instructions for graphing and solving systems of equations. To obtain the precise solution, you may need to employ algebraic methods or use software that can accurately graph and determine the coordinates of the intersection point.