Answer :
Let's analyze the given system of equations to determine the relationship between the two lines it represents:
Equation 1: [tex]\( x - y = 1 \)[/tex]
To describe the system in terms of the slope (m) and y-intercept (b), we want to rearrange the equation into the slope-intercept form: [tex]\( y = mx + b \)[/tex].
Let's solve for y:
[tex]\[ x - y = 1 \][/tex]
[tex]\[ -y = 1 - x \][/tex] (Subtract x from both sides)
[tex]\[ y = x - 1 \][/tex] (Multiply both sides by -1)
Now we have the first equation in slope-intercept form: [tex]\( y = x - 1 \)[/tex]. From this form, we can see that the slope (m) is 1 and the y-intercept (b) is -1.
Equation 2: [tex]\( x = y + 1 \)[/tex]
This equation is also already in a form where we can identify the slope and y-intercept. However, to make it look like the standard slope-intercept form, we can write it as:
[tex]\[ y = x - 1 \][/tex]
Just like the first equation, the second equation also has a slope (m) of 1 and a y-intercept (b) of -1.
Now, since both equations have the same slope and the same y-intercept, they represent the same line on a graph. There will not be any intersection points other than all points on the line itself, as both equations are essentially identical.
Therefore, the correct description for the system of equations is that the lines are the same (coincident), with a positive slope (m > 0). This corresponds to option B: Same lines, m > 0.
Equation 1: [tex]\( x - y = 1 \)[/tex]
To describe the system in terms of the slope (m) and y-intercept (b), we want to rearrange the equation into the slope-intercept form: [tex]\( y = mx + b \)[/tex].
Let's solve for y:
[tex]\[ x - y = 1 \][/tex]
[tex]\[ -y = 1 - x \][/tex] (Subtract x from both sides)
[tex]\[ y = x - 1 \][/tex] (Multiply both sides by -1)
Now we have the first equation in slope-intercept form: [tex]\( y = x - 1 \)[/tex]. From this form, we can see that the slope (m) is 1 and the y-intercept (b) is -1.
Equation 2: [tex]\( x = y + 1 \)[/tex]
This equation is also already in a form where we can identify the slope and y-intercept. However, to make it look like the standard slope-intercept form, we can write it as:
[tex]\[ y = x - 1 \][/tex]
Just like the first equation, the second equation also has a slope (m) of 1 and a y-intercept (b) of -1.
Now, since both equations have the same slope and the same y-intercept, they represent the same line on a graph. There will not be any intersection points other than all points on the line itself, as both equations are essentially identical.
Therefore, the correct description for the system of equations is that the lines are the same (coincident), with a positive slope (m > 0). This corresponds to option B: Same lines, m > 0.
