Solve the system of equations by graphing Then classify the system as consistent or inconsistent and dependent or independent
To solve a system of equations by graphing, you will need to graph the equations on the same coordinate plane and find the point(s) of intersection, if any. Here's a step-by-step explanation:
Step 1: Write down the given equations in standard form. For example, let's consider the system of equations:
Equation 1: 2x + y = 4
Equation 2: 3x - y = 2
Step 2: Pick a range of values for x and substitute them into each equation to find the corresponding y-values. You can choose any range that allows you to see the general shape and location of the lines on the graph.
Let's choose -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5 as values for x.
For Equation 1:
When x = -5, y = 14
When x = -4, y = 12
When x = -3, y = 10
When x = -2, y = 8
When x = -1, y = 6
When x = 0, y = 4
When x = 1, y = 2
When x = 2, y = 0
When x = 3, y = -2
When x = 4, y = -4
When x = 5, y = -6
For Equation 2:
When x = -5, y = -17
When x = -4, y = -14
When x = -3, y = -11
When x = -2, y = -8
When x = -1, y = -5
When x = 0, y = -2
When x = 1, y = 1
When x = 2, y = 4
When x = 3, y = 7
When x = 4, y = 10
When x = 5, y = 13
Step 3: Plot the points obtained for each equation on a graph. Remember to label the axes and scale them appropriately to accommodate the coordinates.
Step 4: Connect the plotted points for each equation with a straight line. Make sure the lines extend beyond the range of values you chose for x to ensure accurate intersection.
Step 5: Determine the point(s) of intersection, if any. The point(s) where the lines intersect is the solution to the system of equations.
Step 6: Now, classify the system as consistent or inconsistent and dependent or independent based on the graph.
If the lines intersect at a single point, the system is consistent and independent. This means there is a unique solution to the system.
If the lines are coincident (meaning they lie on top of each other) and overlap throughout, the system is consistent and dependent. This indicates that the equations are essentially the same and have infinitely many solutions.
If the lines are parallel, they will never intersect, meaning the system is inconsistent. In this case, there is no solution to the system.
By examining the graph with the lines you have drawn, you should be able to classify the system as consistent or inconsistent and dependent or independent based on the intersection, coincidence, or parallel nature of the lines.