Solving the system y = −x +3 and x + y = 8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
This statement is false.
To solve the system of equations, y = -x + 3 and x + y = 8, we can graph the two equations on the same coordinate plane and find the point where they intersect, which represents the solution to the system.
Graphing the equation y = -x + 3 gives a straight line with a slope of -1 and a y-intercept of 3.
Graphing the equation x + y = 8 gives a straight line with a slope of -1 and a y-intercept of 8.
If we graph the two equations on the same coordinate plane, we find that the lines intersect at the point (2, 6). Therefore, there is a solution to the system, and the statement is false.
Does the graph show the system of equations x + y = 2 and −x + y = 7? Should the lines for the system be parallel?
No, the graph does not show the system of equations x + y = 2 and -x + y = 7. The correct system of equations should be x + y = 2 and -x + y = 7.
If we simplify the second equation, we get y = x + 7. This equation has a positive slope of 1, meaning that the line will go up as we move to the right. In contrast, the first equation y = -x + 2 has a negative slope of -1, meaning that the line will go down as we move to the right.
Since the slopes of the two lines are different, they cannot be parallel.
To determine whether the statement is true or false, we need to graph the system of equations and see if they intersect at any point. If they intersect, then there is a solution; if they do not intersect, then there is no solution.
Let's start by graphing the first equation, y = -x + 3. To do this, plot a few points on the graph and then connect them with a straight line. Recall that the equation is in slope-intercept form (y = mx + b), so the y-intercept is 3, and the slope is -1.
Now, let's graph the second equation, x + y = 8. We can rewrite it in slope-intercept form by solving for y: y = -x + 8. Again, plot some points and connect them with a line.
After graphing both equations, we can see that the lines are parallel and never intersect. Therefore, there is no solution to the system. So, the statement "there is no solution to the system" is true.
In conclusion, to solve the system of equations by graphing, we plot the lines for each equation and determine if they intersect or not. If they do not intersect, there is no solution, as with this example.