Solve the system of equations by graphing. Then classify the system?
y= -x -11
4x -3y=-2
What is the solution of the system?
(Type an ordered pair. Type N if there is no solution. Type R if the solution is all real numbers.)
To solve the system of equations by graphing, we need to graph both equations on the same coordinate plane and identify the point where the lines intersect, if they do. This point of intersection will represent the solution to the system of equations.
1. Graphing the first equation, y = -x - 11:
- Start by plotting the y-intercept at (0, -11).
- Then, use the slope of -1/1 to find another point. Since the coefficient of x is -1, we move down 1 unit and right 1 unit from the y-intercept. Plot this point at (1, -12).
2. Graphing the second equation, 4x - 3y = -2:
- To make it easier to graph, we can rewrite the equation in slope-intercept form, y = mx + b. In this case, we solve for y and get y = (4/3)x + (2/3).
- Start by plotting the y-intercept at (0, 2/3).
- Then, use the slope of 4/3 to find another point. Since the coefficient of x is 4/3, we move up 4 units and right 3 units from the y-intercept. Plot this point at (3, 6/3) or (3, 2).
3. Now, on the same coordinate plane, plot both lines and see where they intersect. If they intersect at a single point, that point represents the solution to the system of equations. If the lines are parallel and do not intersect, there is no solution. If the lines coincide or overlap, the solution is all real numbers.
In this case, by graphing the equations, we can see that the lines intersect at the point (-7, 4).
Therefore, the solution to the system of equations is (-7, 4).
The classification of the system of equations is that it has a unique solution.
To solve the system of equations by graphing, we need to graph each equation and find the point(s) where the two lines intersect.
First, let's graph the equation y = -x - 11:
To graph this equation, we can start by finding two points that satisfy the equation. Let's choose x = 0 and x = -5:
When x = 0, y = -0 - 11 = -11. So one point on the graph is (0, -11).
When x = -5, y = -(-5) - 11 = 5 - 11 = -6. Another point on the graph is (-5, -6).
Plot these two points and draw a line passing through them:
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-20----------------------
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-10----------------------
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0-----------------------
| *
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10-----------------------
Next, let's graph the second equation 4x - 3y = -2:
To graph this equation, we use a similar process. Choose two values for x, and solve for y to find two points on the line.
When x = 0, 4(0) - 3y = -2, which gives -3y = -2 and therefore y = 2/3. So one point on the graph is (0, 2/3).
When x = -5, 4(-5) - 3y = -2. Simplifying this equation gives -20 - 3y = -2, which leads to -3y = 18, and therefore y = -18/3 = -6. Another point on the graph is (-5, -6).
Plot these two points and draw a line passing through them:
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-20----------------------
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-10----------------------
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0-----------------------
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10-----------------------
Now, we can see that the two lines intersect at the point (-4, -7). Therefore, the solution to the system of equations is (-4, -7).
So, the solution of the system is (-4, -7).