Which of these graphs show that the linear system Y equals negative X +6 and 3X plus 3Y equals 18 has an infinite number of solutions
To determine which of the graphs show that the given linear system has an infinite number of solutions, let's first rewrite the equations in slope-intercept form (Y = mx + b).
1. Y = -X + 6
2. 3X + 3Y = 18
To convert the second equation into slope-intercept form, we need to solve for Y:
3X + 3Y = 18
3Y = -3X + 18
Y = -X + 6
Comparing the equations in slope-intercept form, we can see that both equations have the same slope (-1), and the y-intercepts are also the same (6). This means that the lines represented by the equations are identical and will intersect at every point. Therefore, the given linear system has an infinite number of solutions.
Now, let's analyze the graphs provided and identify the one that matches the given condition.
To determine which graph shows that the linear system has an infinite number of solutions, we need to first rewrite the equations in slope-intercept form (Y = mx + b).
Equation 1: Y = -X + 6
Equation 2: 3X + 3Y = 18
Now, let's analyze the equations and look for any similarities or relationships between them.
In Equation 1, the slope (m) is -1, meaning that for every increase of 1 in X, Y decreases by 1. The y-intercept (b) is 6, indicating that the line crosses the y-axis at (0, 6).
In Equation 2, we need to manipulate the equation to get it into slope-intercept form. First, we divide both sides of the equation by 3 to simplify it:
1X + 1Y = 6
Now we subtract X from both sides to isolate the Y term:
1Y = -1X + 6
Simplifying further, we divide both sides by 1:
Y = -1X + 6
Comparing both equations, we can see that they are equivalent. The equations have the same slope and y-intercept, indicating that they represent the same line. Because both equations are different forms of the same line, it means that the system has an infinite number of solutions.
Therefore, any graph where the lines formed by the equations overlap entirely, intersect in every point, or are coincident represents a system with an infinite number of solutions.