Final answer:
Part A: (0,3) is a solution to Line 1. Part B: False. Part C: The slopes of Line 1 and Line 2 are 2 and -3, respectively. Part D: The y-intercepts of Line 1 and Line 2 are 3 and -3, respectively. Part E: A graph of Line 1 and Line 2 can be sketched. Part F: There is no solution to the system of equations represented by Line 1 and Line 2.
Step-by-step explanation:
Part A: To determine if (0,3) is a solution to Line 1, we substitute x=0 and y=3 into the equation and check if the equation is satisfied. Substituting the values, we get:
2(0) - 3 = -3
Simplifying, we have:
-3 = -3
Since the equation is true, (0,3) is a solution to Line 1.
Part B: To determine if (0,-3) is a solution to Line 2, we substitute x=0 and y=-3 into the equation and check if the equation is satisfied. Substituting the values, we get:
-6(0) - 2(-3) = -6
Simplifying, we have:
6 = -6
Since the equation is false, (0,-3) is not a solution to Line 2.
Part C: The slope of Line 1 can be determined by rearranging the equation into the slope-intercept form y=mx+b, where m is the slope. Rearranging Line 1, we have:
y = 2x + 3
Comparing this equation to the slope-intercept form, we can see that the slope of Line 1 is 2. Similarly, the slope of Line 2 can be determined by rearranging the equation into the slope-intercept form. Rearranging Line 2, we have:
y = -3x + 6
Comparing this equation to the slope-intercept form, we can see that the slope of Line 2 is -3.
Part D: The y-intercept of Line 1 is the value of y when x=0. From the equation, we have:
2(0) - y = -3
Simplifying, we get:
-y = -3
So the y-intercept of Line 1 is 3. Similarly, the y-intercept of Line 2 is the value of y when x=0. From the equation, we have:
-6(0) - 2y = -6
Simplifying, we get:
-2y = -6
So the y-intercept of Line 2 is -3.
Part E: To sketch the graph of Line 1 and Line 2, we can use the slope-intercept form. For Line 1, we know the slope is 2 and the y-intercept is 3, so we can plot the point (0,3) and use the slope to find additional points. For Line 2, we know the slope is -3 and the y-intercept is -3, so we can plot the point (0,-3) and use the slope to find additional points. Finally, we can draw a line through the points to represent each line.
Part F: The solution to the system of equations represented by Line 1 and Line 2 is the point where the two lines intersect. From the graph, it appears that the lines do not intersect, so there is no solution to the system of equations.