Solve each system graphically. Be sure to check your solution. If a system has an infinite number of solutions, use set−builder notation to write the solution set. If a system has no solution, state this. 6x – 2y = 2, 9x – 3y = 1
"Solve each system graphically"
You'll need to graph each equation and look for the intersection.
If there is no intersection, the system has no solution.
If the two lines coincide, the system has infinite solutions.
If the two lines intersection at one point, there is a unique solution.
Try working it out on your graphics calculator or your school's math package.
To solve the given system of equations graphically, we need to plot the lines represented by each equation on a coordinate plane and find their point of intersection.
Let's start with the first equation: 6x - 2y = 2. To graph this equation, we can rearrange it into slope-intercept form (y = mx + b) by isolating y:
-2y = -6x + 2
Dividing all terms by -2:
y = 3x - 1/2
Now let's move on to the second equation: 9x - 3y = 1. Again, we can rearrange it into slope-intercept form:
-3y = -9x + 1
Dividing all terms by -3:
y = 3x - 1/3
Now that we have both equations in slope-intercept form, we can graph them.
Graph the first equation, y = 3x - 1/2:
1. Start by plotting the y-intercept, which is -1/2, at the point (0, -1/2).
2. Determine the slope, which is 3. This means that for every increase of 1 in x, y increases by 3 units. Plot additional points using this information.
3. Draw a straight line passing through the plotted points. This line represents the first equation.
Graph the second equation, y = 3x - 1/3:
1. Start by plotting the y-intercept, which is -1/3, at the point (0, -1/3).
2. Determine the slope, which is also 3. Plot additional points using this information.
3. Draw a straight line passing through the plotted points. This line represents the second equation.
Now that we have graphed both equations, we can visually observe the point(s) of intersection between the lines. This point represents the solution to the system of equations.
Upon observation, we find that the two lines are coincident. In other words, they overlap completely. This tells us that the system has an infinite number of solutions.
To express the solution set in set-builder notation, we can write it as:
{(x, y) | x is a real number, y = 3x - 1/2}
To check the solution, substitute any value of x into either of the original equations and solve for y. You will find that the equation is satisfied for any value of x, confirming that the solution set is infinite.