solve each system graphically. be sure to check your solution. if a system has an infinite number of solutions, use the set-builder notation to write solution set. if a system has no solution, state this. where appropriate, round to the nearest hundredth.
6x-2y=2
9x-3y=1
since the lines have the same slope, they are parallel. If you graph them, they will not intersect.
what does that say about the number of solutions?
To solve the system of equations graphically, we will plot the equations on a coordinate plane and determine the point(s) where they intersect. Let's graph the equations:
Equation 1: 6x - 2y = 2
Equation 2: 9x - 3y = 1
Step 1: Rearrange the equations in slope-intercept form (y = mx + b) to easily identify the slope and y-intercept.
Equation 1: -2y = -6x + 2
y = 3x - 1
Equation 2: -3y = -9x + 1
y = 3x - 1/3
Step 2: Plot the lines on a coordinate plane using the slopes and y-intercepts.
For Equation 1, when x = 0, y = -1. So we have the point (0, -1). From this point, we can identify another point by using the slope. The slope is 3, meaning for every increase of 1 in the x-coordinate, there is an increase of 3 in the y-coordinate. Therefore, the next point can be (1,2). Plot these two points and draw a line passing through them.
For Equation 2, when x = 0, y = -1/3. So we have the point (0, -1/3). Using the slope of 3, we can find another point. Plot the point (1, 2/3) and draw the line passing through both points.
Step 3: Determine the intersection point(s) of the lines.
By visually examining the graph, we can see that the two lines intersect at the point (1, 2/3).
Step 4: Check the solution.
Substitute the x-coordinate and y-coordinate of the intersection point into both equations to confirm if they satisfy the system.
For Equation 1: 6(1) - 2(2/3) = 6 - 4/3 = 18/3 - 4/3 = 14/3 ≈ 4.67
For Equation 2: 9(1) - 3(2/3) = 9 - 2 = 7
Both equations yield the same values, so the point (1, 2/3) is indeed a solution to the system of equations.
Therefore, the system has a unique solution: (1, 2/3).
Note: If the lines were coincident (falling on top of each other), the system would have infinitely many solutions. If the lines were parallel, the system would have no solution.