Solve the system of equations by graphing. Then clasify the the system.
7x-9y=-54
9y-7x=54
Those equations are not independent. They will plot as the same line, and one can be derived from the other. There are an infinite number of solutions.
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To solve the system of equations by graphing, we first need to rewrite the given equations in slope-intercept form, which is of the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
1. Equation 1: 7x - 9y = -54
Subtract 7x from both sides of the equation:
-9y = -7x - 54
Divide both sides of the equation by -9:
y = (7/9)x + 6
2. Equation 2: 9y - 7x = 54
Add 7x to both sides of the equation:
9y = 7x + 54
Divide both sides of the equation by 9:
y = (7/9)x + 6
Now that we have both equations in slope-intercept form, we can graph them on the same coordinate plane.
1. Start by drawing the x and y axes on a graph paper.
2. Locate the y-intercept of each equation on the y-axis.
For Equation 1:
The y-intercept is 6, so plot the point (0, 6).
For Equation 2:
The y-intercept is 6, so plot the point (0, 6) again.
3. Now, use the slope of each equation to find a second point for each line. The slope of both equations is 7/9, which means that for every increase of 9 units in the x-direction, the y-value increases by 7 units.
For Equation 1:
From the y-intercept (0, 6), move up 7 units and to the right 9 units. Plot the point (9, 13).
For Equation 2:
From the y-intercept (0, 6), move up 7 units and to the right 9 units. Plot the point (9, 13).
4. Connect the points for each equation using a straight line. Equation 1 will be a line passing through the points (0, 6) and (9, 13). Equation 2 will also be a line passing through the points (0, 6) and (9, 13).
After graphing the lines, we can visually determine the solution to the system of equations.
To classify the system, we need to analyze the graph of the lines:
- If the lines intersect at a single point, the system has one solution, and the equations are considered independent.
- If the lines are parallel and do not intersect, the system has no solution, and the equations are considered inconsistent.
- If the lines overlap and have infinitely many points of intersection, the system has infinitely many solutions, and the equations are considered dependent.
By looking at the graph, we can observe that the lines are identical and perfectly overlap. Hence, the system of equations has infinitely many solutions, and the equations are dependent.
Note: While graphing is one method to solve a system of equations, it is beneficial to also learn algebraic methods such as substitution or elimination. These methods give precise solutions and can be used efficiently for any system of equations.