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Which of these graphs shows that the linear system y = -x + 6 and 3x + 3y = 18 has an infinite number of solutions? (1 point) O O minu 104 9 -8-4 7 6 5 3- 2 -14 -04 -10-9-8-7-6-5-4-3-2-1,0 1 2 3 -24 -3- -4 -5- -64 -7- -84 -9- -10- 10- 9 8 7 6 5. 4 3 24 -1 -0+ -10-9-8-7-6-5-4-3-2-1,0 2 3 4 4 5 6 7 8 9 10 -11 -2- 4 5 6 7 8 9 10 -3- -4- -5- -6- -7 -8- -9- -104
The graph that shows that the linear system y = -x + 6 and 3x + 3y = 18 has an infinite number of solutions is the first graph, which is a straight line with a negative slope passing through the point (0, 6).
To determine which graph represents a linear system with an infinite number of solutions, we need to analyze the equations and their graphical representations.
The given linear system consists of two equations:
1. y = -x + 6
2. 3x + 3y = 18
To find the graph that represents the system with an infinite number of solutions, we need to look for two equations that are equivalent. We can do this by rearranging the second equation so that it is in the same form as the first equation.
Let's start by rearranging the second equation:
3x + 3y = 18
Dividing by 3 on both sides, we get:
x + y = 6
Now we can see that the two equations are the same. Both equations represent the same line: y = -x + 6.
Looking at the provided graphs, option O on the leftmost side depicts the line y = -x + 6. Since both equations are equivalent, this means that the system has an infinite number of solutions.
Therefore, the correct answer is option O.
To determine which graph shows that the linear system y = -x + 6 and 3x + 3y = 18 has an infinite number of solutions, we need to find the graph where the two lines overlap or are coincident.
Here's how to find the solution using the given equations:
1. Start with the first equation: y = -x + 6
- Rearrange it to slope-intercept form (y = mx + b): y = -1x + 6
- Slope (m) is -1 and y-intercept (b) is 6.
2. Move to the second equation: 3x + 3y = 18
- Divide all the terms by 3 to simplify: x + y = 6
- Rearrange it to slope-intercept form: y = -1x + 6
- Here again, the slope is -1 and y-intercept is 6.
Since the equations have the same slope and y-intercept, it means they represent the same line. This implies that the system has an infinite number of solutions since the lines are coincident or overlap.
Now, examining the provided answer choices:
The answer choice that matches this condition is the graph that shows a single line passing through the points (-1, 7), (0, 6), and (1, 5). Therefore, the correct graph is the one that appears to be a straight line passing through these points.
If you can provide further information about the answer choices or clarify any additional points of confusion, feel free to do so.