Solve the following pair of simultaneously equation graphically using gradient method
X+2y=3
2x+y=0
To solve the given system of equations graphically, we need to plot the lines represented by each equation and find the point of intersection (if any).
1. Observe the first equation: x + 2y = 3.
To plot this line, rearrange the equation in terms of y:
2y = 3 - x
y = (3 - x)/2
Now, choose some values for x and calculate the corresponding values of y:
For x = 0, y = (3 - 0)/2 = 1.5
For x = 3, y = (3 - 3)/2 = 0
Plot the points (0, 1.5) and (3, 0) on the graph.
2. Observe the second equation: 2x + y = 0.
To plot this line, rearrange the equation in terms of y:
y = -2x
Choose some values for x and calculate the corresponding values of y:
For x = 0, y = 0
For x = 1, y = -2
Plot the points (0, 0) and (1, -2) on the graph.
Now, plot the two lines on the same graph and find their point of intersection, if any. Here is the graph:
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The two lines intersect at the point (1, -1). Thus, the solution to the system of equations is x = 1 and y = -1.
To solve the pair of simultaneous equations graphically using the gradient method, we need to plot the graphs of both equations, where the point of intersection corresponds to the solution.
1. Convert the equations into slope-intercept form (y = mx + b):
Equation 1: x + 2y = 3
Rewrite as: y = (-1/2)x + 3/2
Equation 2: 2x + y = 0
Rewrite as: y = -2x
2. Plot the graphs:
- Start by plotting the first equation, y = (-1/2)x + 3/2.
- At x = 0, y = 3/2: plot the point (0, 3/2).
- For the second point, x can be any value. Let's choose x = 2.
Plugging x = 2 into the equation, we get y = -2+3/2 = -1/2. Plot the point (2, -1/2).
- Draw a straight line passing through these two points.
- Next, plot the second equation, y = -2x.
- Start with x = 0, y = 0. Plot the point (0, 0).
- For the second point, let's choose x = 2.
Plugging x = 2 into the equation, we get y = -2(2) = -4. Plot the point (2, -4).
- Draw a straight line passing through these two points.
3. Find the point of intersection:
The point where the two lines intersect is the solution to the simultaneous equations.
- Based on the graph, we can observe that the two lines intersect at the point (2, -1).
Therefore, the solution to the simultaneous equations is x = 2 and y = -1.
To solve the pair of simultaneous equations graphically using the gradient method, we can follow these steps:
Step 1: Rearrange each equation in the form y = mx + c, where m represents the gradient and c represents the y-intercept.
Equation 1: x + 2y = 3
Rearranging, we get 2y = -x + 3, and dividing by 2, y = -1/2x + 3/2
Equation 2: 2x + y = 0
Rearranging, we get y = -2x
Step 2: Plot the graphs of both equations on the same set of axes.
For Equation 1, start by plotting the y-intercept, which is (0, 3/2). Then, using the slope (-1/2), find another point by moving one unit to the right and two units down. For simplicity, plot two additional points - one to the left of the y-intercept and one to the right. Connect the points to form a straight line.
For Equation 2, start by plotting the y-intercept, which is (0, 0). Then, using the slope (-2), find another point by moving one unit to the right and two units down. Plot two additional points - one to the left of the y-intercept and one to the right. Connect the points to form a straight line.
Step 3: Determine the point of intersection.
Find the point where the two lines intersect. This point represents the solution to the simultaneous equations.
Step 4: Read the coordinates of the point of intersection.
Once you have identified the point of intersection, read the coordinates of that point. The x-coordinate will give you the value of x, and the y-coordinate will give you the value of y, which is the solution to the simultaneous equations.
That's how you can solve the pair of simultaneous equations graphically using the gradient method.